JON 0. NEWMAN, Circuit Judge:

This appeal in a criminal case poses, in a novel context, the issue of what options are open to a defendant and a sentencing judge on remand when a sentence exceeds a sentence bargain that specified only one component of the sentence. The issue arises on an appeal by Edward Bohn and Maxine Heckroth from judgments entered. July 12, 1991, in the District Court for the Eastern District of New York (Thomas C. Platt, Chief Judge), convicting them on their pleas of guilty to mail fraud, in violation of 18 U.S.C. § 1341 (1988), and, in Bohn’s case, also to money laundering, in violation of 31 U.S.C. § 5324(3) (1988). We conclude that the District Judge must either permit withdrawal of the guilty pleas or conform his sentence to the sentence bargains; that, if the pleas stand, the sentencing judge has discretion to increase the components of the sentence that were not included in the sentence bargains; and that the defendants have an additional option to withdraw this appeal now that they are aware that their appellate “victory” risks consequences that they might well regard as adverse.

Facts

Both appellants were charged with mail fraud offenses and Bohn was charged with money laundering offenses arising out of their telemarketing of computer office supplies. Both pled guilty on October 5, 1990, pursuant to plea bargains. Bohn agreed to plead guilty to one count of mail fraud and one count of money laundering, and Heck-roth agreed to plead guilty to two counts of mail fraud. Both defendants agreed to cooperate with the Government in its ongoing investigation, and the Government agreed to make a motion for a cooperation *391 departure under section 5K1.1 of the Sentencing Guidelines. An unusual aspect of the plea agreements and the one out of which these appeals arise is that each agreement specified the amount of the fine but were silent on all other aspects of the sentence. See Fed.R.Crim.P. 11(e)(1)(C). Bohn’s agreement specified a fine of $125,-000, and Heckroth’s agreement specified a fine of $50,000. Both agreements specified these fine amounts, but neither contained any language limiting the sentence to a fine. Thus, both agreements permitted the sentencing judge to include a term of imprisonment.

At defendants’ sentencing on June 28, 1991, the District Judge asked the prosecutor whether he recommended probation. The prosecutor replied that he had no objection to probation. Chief Judge Platt said he understood that reply to be a recommendation for probation. The Judge sentenced Bohn to five years’ probation and a fine of $500,000. He specifically stated that he was going to impose a fine “in light of the fact that I am not going to sentence you to jail.” The Judge sentenced Heckroth to five years’ probation and a fine of $250,000. In both cases, the fines exceeded the amounts specified in the sentence bargains.

Discussion

On appeal, the defendants and the Government are in agreement that both judgments must be remanded for resen-tencing and that, on remand, the District Judge should be directed to reduce the fines to the amounts specified in the sentence bargains. The parties disagree, however, as to what should happen to the remainder of the judgments. The defendants contend that the remainder of each judgment, notably the component imposing a sentence of probation, should not be changed. The Government contends that the District Judge should have the option of replacing the sentences of probation with terms of imprisonment, up to the statutory máximums. These conflicting contentions frame the principal issue, but do not exhaust the complexities of this appeal.

We begin our analysis, as the parties do, with the undisputed fact that as to both defendants the fines imposed exceed the sums specified in the sentence bargains. When a plea bargain, or in this case, a sentence bargain, has not been honored, the remedy is either to enforce the bargain or to afford the defendant an opportunity to vacate the guilty plea. Santobello v. New York, 404 U.S. 257, 92 S.Ct. 495, 30 L.Ed.2d 427 (1971). It is not entirely clear who is entitled to exercise the Santobello option — the court or the defendant. See id. at 263, 92 S.Ct. at 499 (leaving choice to state court on remand); United States v. Kurkculer, 918 F.2d 295, 298-302 (1st Cir.1990) (remanding for imposition of specified sentence); United States v. Burruezo, 704 F.2d 33, 38-39 (2d Cir.1983) (requiring opportunity to withdraw plea, where such relief unopposed by prosecution). We think that initially the option lies with the District Judge, at least in a case such as this that does not involve lack of compliance by the prosecution. Thus, on remand, Chief Judge Platt must initially determine whether he wishes to accept the sentence bargains. If he accepts the bargain in either defendant’s case, he must conform the sentence to that bargain by reducing the fine to the bargained amount. If, instead, he prefers in either defendant’s case to retain the authority to impose a fine greater than the bargained amount, then he must afford that defendant the opportunity to withdraw the guilty plea.

The Government contends that only the option of enforcing the sentence bargains should now be available because the Government would be prejudiced by having to reassemble its trial evidence more than a year after the guilty pleas were entered. We think the Government has lost the opportunity to make that complaint in this Court by virtue of its failure to alert the District Judge to the fact that his sentences exceeded the sentence bargains. Had the Government pointed this out at sentencing, the District Judge could then have determined whether to adhere to the sentence bargains or to permit the defendants to withdraw their pleas. The *392 Judge’s sentencing options should not be restricted by a sentence bargain he is entitled to reject, see Fed.R.Crim.P. 11(e)(4), just because the Government neglected to call his attention to a provision of the bargain. Of course, the Government is free to argue to the District Judge that the burdens it faces in reassembling its trial evidence weigh in favor of a decision by the Judge to exercise his discretion to accept the sentence bargains and conform his sentences to their terms.

The more substantial issue is whether on remand the sentencing judge, if he elects to accept the sentence bargains, may impose a sentence of imprisonment in lieu of the sentence of probation, once he lowers the fines to the bargained amounts. This issue comprehends several questions.

The first question is whether we should determine the Judge’s authority at this point, or simply remand and consider the issue on a subsequent appeal in the event that he accepts the sentence bargains and elects to impose sentences of imprisonment. Though withholding rulings until events require them is normally sensible appellate practice, it would be unfair to the defendants to follow that course in this case. If the Judge has the option to impose prison sentences, the defendants are entitled to know that before they have irrevocably placed themselves in a position where their appellate “victory” in challenging the fines exposes them to the prospect of adverse consequences.

The second question is whether the Government may be heard in support of the option of imprisonment without having taken a cross-appeal. It is a familiar rule that an “appellee may not attack the decree with a view either to enlarging his own rights thereunder or of lessening the rights of his adversary, whether what he seeks is to correct an error or to supplement the decree with respect to a matter not dealt with below.” United States v. American Railway Express Co., 265 U.S. 425, 435, 44 S.Ct. 560, 564, 68 L.Ed. 1087 (1924). An appellee who wishes to augment a judgment in some respect is normally obliged to file a cross-appeal and proceed as a cross-appellant. See 9 Moore’s Federal Practice, ¶ 204.11[3] (1991).

Whether a cross-appeal is required where an appellee wishes a component of the judgment increased only in the event that another component is decreased on appeal is a matter of some uncertainty and surprisingly little authority. See Hartford Accident and Indemnity Co. v. Sullivan, 846 F.2d 377, 385 (7th Cir.1988) (Posner, J.) (whether cross-appeal rule applies “where the modification is conditional on the appellate court’s rejecting other parts of the district court’s decision is a question on which we can find no authority”), cert. denied, 490 U.S. 1089, 109 S.Ct. 2428, 104 L.Ed.2d 985 (1989). Three decisions of the Supreme Court, all in the civil context, shed some light on the problem. In Alexander v. Cosden Pipe Line Co., 290 U.S. 484, 487, 54 S.Ct. 292, 293, 78 L.Ed. 452 (1934), the Court ruled that, at least in Supreme Court practice, a party that had suffered adverse outcomes in the Court of Appeals on two distinct portions of a judgment in a suit for wrongful collection of taxes could not, in the absence of a cross-petition for certiora-ri, obtain review of those adverse rulings in response to a successful effort by the tax collector to increase two other portions of the judgment. In Helvering v. Pfeiffer, 302 U.S. 247, 250-51, 58 S.Ct. 159, 160-61, 82 L.Ed. 231 (1937), the Court applied the Cosden approach to bar the Commissioner of Internal Revenue from seeking, without a cross-appeal, an increase in one portion of a deficiency assessment in response to a ruling by the Court of Appeals decreasing another portion of the same year’s assessment.

More recently, however, the Court has given contrary signals, at least with respect to closely related portions of a judgment. In Bowen v. United States Postal Service, 459 U.S. 212, 103 S.Ct. 588, 74 L.Ed.2d 402 (1983), a plaintiff obtained a judgment of $30,000 against a union and $22,954 against an employer, representing their apportioned shares of a back pay award. The Court of Appeals reversed the judgment against the Union and refused to increase the judgment against the employ *393 er for lack of a cross-appeal by the plaintiff-appellee. The Supreme Court reversed the Court of Appeals, reinstating the District Court’s allocation. The Supreme Court majority noted in dictum that the Court of Appeals had erred in ruling that the lack of a conditional cross-appeal would have precluded an increase in the judgment against the employer had the rejection of the judgment against the union remained undisturbed. 459 U.S. at 217-18 n. 7, 103 S.Ct. at 592-93 n. 7. Three justices, who dissented in part on the ground that the award against the union should have been rejected, voted to increase the award against the employer, notwithstanding the absence of a conditional cross-appeal. 459 U.S. at 244-45, 103 S.Ct. at 606-07 (White, J., with whom Marshall and Blackmun, JJ., join, dissenting in part). Only then-Justice Rehnquist expressed the view that the absence of a conditional cross-appeal would have precluded the plaintiff from obtaining an increase in the judgment against the employer, if the judgment against the union had been rejected, as he thought it should have been. Id. at 246-47, 103 S.Ct. at 607-08 (Rehnquist, J., dissenting). The willingness of eight members of the Court to approve an increase in the award against the employer without a cross-appeal conditioned on a decrease in the award against the union suggests that the Cosden/Pfeif-fer approach is no longer to be strictly followed, at least with respect to components of a judgment as closely related as two components of an apportioned back pay award.

Even more uncertainty attends the application of the cross-appeal requirement to the criminal context, a circumstance likely arising from the restricted opportunity the Government has to become an appellant in criminal cases. With the recent expansion of Government appellate rights with respect to criminal sentences, see 18 U.S.C. § 3742(b) (1988), the cross-appeal question can be expected to recur more frequently. In the past, without pausing to consider the significance of the lack of a conditional cross-appeal, we have entertained Government requests to augment one component of a sentence in response to appellate rulings decreasing another component where the aggregate sentence would not be increased. In United States v. Diaz, 778 F.2d 86, 88-89 (2d Cir.1985), we permitted the Government to argue, successfully, that on remand from the invalidation of a mandatory consecutive sentence, the sentencing judge should have the option of increasing the term imposed on a remaining valid count, i.e., the term previously imposed “underneath” the invalidated sentence that the judge had thought was required. See McClain v. United States, 643 F.2d 911, 913-14 (2d Cir.1981) (remanded to permit Government to argue to District Court that the option, later recognized in Diaz, should be available with respect to sentence “underneath” invalidated mandatory sentence); McClain v. United States, 676 F.2d 915, 917-19 (2d Cir.) (approving District Court’s exercise of option to increase sentence), cert. denied, 459 U.S. 879, 103 S.Ct. 174, 74 L.Ed.2d 143 (1982). In United States v. Pisani, 787 F.2d 71, 75-76 (2d Cir.1986), we entertained but rejected on the merits the Government’s argument that after invalidation of a sentence on one count, the sentencing judge should have the option of increasing the sentence on another, unrelated count. In United States v. Young, 932 F.2d 1035, 1037-38 (2d Cir.1991), we permitted the Government to argue, successfully, that on remand from the invalidation of a restitution order, the sentencing judge, who had explicitly withheld a fine because of the restitution order, should have the option of imposing a fine. Most recently, in United States v. Gelb, 944 F.2d 52, 58-60 (2d Cir.1991), we permitted the Government to argue, successfully, that on remand from the invalidation of sentences on tax counts covering certain taxable years, the sentencing judge should have the option of increasing the sentence on an identical tax count for a different, though consecutive, year. In none of these cases had the Government taken a conditional cross-appeal.

In light of Bowen and the above-cited group of criminal cases in this Circuit, we conclude that a conditional cross-appeal is not required, at least in a case like Bohn’s, *394 to permit us to entertain a Government request to augment one component of a sentence on one count in response to an appellate ruling decreasing another component of the sentence on the same count. Requiring conditional cross-appeals in such circumstances would oblige the Government (and defendants in response to Government appeals of sentences) to file conditional cross-appeals reflexively, anticipating all possible appellate adjustments of the sentence in advance of knowing precisely the nature of the appellant’s challenge. That would burden appellees (and courts) with no appreciable benefit to appellate practice. See Robert L. Stern, When to Cross-Appeal or Cross-Petition — Certainty or Confusion?, 87 Harv. L.Rev. 763 (1974). Whether this approach should be applied to entertain, without a conditional cross-appeal, claims for adjustments of sentences on unrelated counts is a matter we may leave for another day.

We turn then to the merits of the Government’s contention that on remand the District Court should be entitled to impose imprisonment if it elects to cr Torce the sentence bargain by reducing the amount of the fines. Not every reduction in a component of a criminal sentence entitles the Government to seek an increase in another component. As we ruled in Pisani, for example, the Government cannot obtain an increase in a sentence on one count in response to a reduction in a sentence on an unrelated count. Here, however, we deal with components of a sentence on a single count. The situation is much closer to Young, where we permitted consideration of a fine in response to the invalidation of an order of restitution on the same count.

The defendants contend that our prior decisions in United States v. Coke, 404 F.2d 836 (2d Cir.1968) (in banc), and United States v. Barash, 428 F.2d 328 (2d Cir.1970), ce rt. denied, 401 U.S. 938, 91 S.Ct. 928, 28 L.Ed.2d 217 (1971), preclude a revision of the sentences to include imprisonment. Coke, anticipating the Supreme Court’s decision in North Carolina v. Pearce, 395 U.S. 711, 89 S.Ct. 2072, 23 L.Ed.2d 656 (1969), held that after a retrial a sentence may not be increased except in the limited circumstances where such an increase is warranted by the defendant’s conduct occurring since the first trial or by information that had not come to the attention of the original sentencing judge as a result of some action of the defendant. Preclusion of an increased sentence in the absence of such special circumstances was thought necessary to avoid impairing the right to challenge a conviction. Barash made clear that Coke applied to bar an increase, after retrial, of any component of punishment, prohibiting a subsequent sentence that included nine months’ imprisonment and fines in place of a prior sentence that included imprisonment for a year and a day but omitted any fines. Appellant attempts to draw from Barash a general rule that, in the absence of the exceptional circumstances noted in Coke, no component of a sentence may be increased even if some other component is decreased, without inquiry as to whether the severity of the aggregate sentence has been increased. The Court in Barash thought it “wholly impractical” to compare one kind of punishment with another and “not possible” to determine whether a fine or a prison term was more severe. 428 F.2d at 332.

Though the matter is not free from doubt, we do not agree that the principle of Coke and Barash, formulated to avoid deterring challenges to a conviction, applies to challenges to a sentence. The increment of punishment sought to be imposed in Coke to the aggregate sentence, and sought to be imposed in Barash to the fine component of the sentence, was totally unrelated to the defendant’s challenge to his conviction. Allowing the increment to stand would have served no penological purpose and might well have had the adverse effect of deterring a challenge to a conviction that was improperly obtained. By contrast, an increase in one component of a sentence, imposed following a successful challenge to another component of a sentence, is intimately related to the exercise of a sentencing judge’s authority to select an appropriate sentence. Though the prospect of increasing one component *395 of the sentence creates an arguable deterrent to a challenge to another component that might be unlawful, we think the legitimate purposes of sentencing counsel against an absolute rule barring all increases of any component after another component has been successfully challenged. The balance between affording the sentencing judge appropriate leeway to refashion a proper sentence and unduly deterring challenges to unlawful components of a sentence can best be struck by sensitive line-drawing between permissible and impermissible resentencings.

We have followed that approach in the series of cases previously cited, permitting in Diaz and McClain a term of imprisonment to be increased after invalidation of a consecutive sentence thought to have been required, permitting in Young a fine to be imposed after invalidation of an order of restitution, and permitting in Gelb a sentence to be increased after invalidation of a sentence on “fungible” tax counts, but prohibiting in Pisani a sentence to be increased on one count after invalidation of a sentence on an unrelated count. We have thus examined carefully the nature of the increase, and, in a proper case, would no doubt also examine the severity of the increase, despite the difficulties identified in Barash.

In the pending case, we think it entirely appropriate to permit the sentencing judge, if he elects to enforce the sentence bargains and reduce the fines, to exercise his discretion to impose terms of imprisonment with respect to the same counts for which the fine component of the sentence will be reduced. The extent of such terms, however, must not be so severe as to create an undue risk of deterring others from subsequent challenges to sentence components that might be unlawful.

According to the sentencing judge an option either to permit withdrawal of the plea, which would eliminate all aspects of the limitations of the sentence bargains, or to let the pleas stand and conform the fines to the sentence bargains, which would expose the defendants to the risk of imprisonment, confronts the appellants with an outcome that they might well regard as less satisfactory than the sentences currently in place. Since appellants could not necessarily have anticipated the risks that their “victory” would entail, we deem it appropriate, in the exercise of our authority to “require such further proceedings to be had as may be just under the circumstances,” 28 U.S.C. § 2106 (1988), to afford them the opportunity to accept their current sentences instead of facing the risk of imprisonment. Accordingly, we will stay our mandate for an additional ten days beyond the normal interval provided by Fed.R.App.P. 41(a), or beyond the disposition of any timely petitions for rehearing, whichever period is longer, during which time either appellant may withdraw his or her appeal.

The sentences of the District Court are vacated, and the cases remanded for further proceedings consistent with this opinion.