The U.S. Supreme Court on Wednesday appeared to find no easy way to resolve an international child custody battle between a U.S. Army soldier and a Scottish woman, a battle with important ramifications for the power of federal courts.

The justices heard arguments in Chafin v. Chafin, in which Sgt. Jeffrey Lee Chafin argues that his appeal of a federal district court order returning his daughter to Scotland is not moot just because the child has left this country—an issue that has created a split in the federal appellate courts.

The Chafin case requires the justices to parse the language and objectives of the Hague Convention on the Civil Aspects of International Child Abduction, to which the United States is a signatory, and their own requirements for determining whether an appeal or case is moot.

The arguments by lawyers for Jeffrey Chafin, now stationed in Virginia, and U.K. citizen Lynne Chafin quickly boiled down to whether it would make any difference if the father got his appeal and won, now that his daughter has been in Scotland with her mother for the last 14 months.

Some justices seemed uncomfortable with the notion that there could be no appeal in these circumstances. Others worried that an appeal could result in competing orders for return of the child and that would undermine the primary objective of the Hague Convention: an end to the back-and-forth transfer of children by the speedy resolution of their “habitual residence.”

The father’s counsel, Michael Manely of The Manely Firm in Marietta, argued that an appeal resulting in reversal of the district court’s judgment that Scotland was the habitual residence of the child under the Hague Convention could give relief to the father in three ways sufficient to preclude mootness.

Reversal would mean that the United States is the child’s habitual residence, said Manely. The district court or court of appeals should then order the return of the child and the reversal of a $94,000 judgment against her father. Finally, he said, it would allow Alabama courts—not Scottish courts—to determine custody.

Manely received a boost from Assistant to the Solicitor General Nicole Saharsky, who told the justices, “Put simply, this case is not moot because an appellate decision on the merits would matter. All the court needs to decide is that the appellate court’s decision would make a difference here. If there is nothing else but a declaratory judgment that the father either did or did not wrongfully retain the child in the U.S., that is a piece of paper in the world that has consequences to these parties. It has consequences in terms of the money judgment that’s been entered, and it has consequences in terms of where custody will be determined.”

But Justice Ruth Bader Ginsburg said, “This child has been in Scotland for 14 months. Now you say bring it back to the United States and we start over. The whole object of the return procedure is so that you get the child to a place that’s a proper place to determine custody.”

The Chafins were married in Scotland and moved to Germany, where he was stationed. In 2007, their daughter was born in Germany and has dual citizenship in the United States and the United Kingdom. When the father was deployed to Afghanistan for 15 months, the Chafins agreed that Lynne Chafin should return to Scotland with her daughter. When his deployment ended, Jeffrey Chafin returned to Germany; Lynne Chafin and her daughter did not. He subsequently was transferred to Alabama in 2009.

Reportedly in an effort to save their marriage, Lynne Chafin, who had a temporary visa, and her daughter went to Alabama in February 2010. But in May, Jeffrey Chafin filed for divorce and an emergency custody order and removed his daughter’s passports to an unknown location. Because she could not leave with her daughter, Lynne Chafin overstayed her visa and was deported in February 2011.

The following May, Lynne Chafin filed a petition for return of her child under the Hague Convention. A federal judge ruled in her favor, finding that Jeffrey Chafin wrongfully retained his daughter by hiding the passports and that Scotland was her habitual residence. The judge rejected a motion to stay the order allowing Lynne Chafin to leave with her daughter, and there was no motion for a temporary stay to allow a motion for a stay in the appellate court. Jeffrey Chafin appealed, and the U.S. Court of Appeals for the Eleventh Circuit dismissed the appeal as moot and remanded the case to the district court to vacate its order and dismiss the case as moot.

Lynne Chafin’s counsel, Stephen Cullen of Washington’s Miles & Stockbridge, told the justices, “The effect that any appeals court could give would be zero in the Scottish court, nothing.”

When Justice Sonia Sotomayor suggested the appellate court could order the return of the child if Jeffrey Chafin won an appeal, Cullen rejected that relief. The Hague convention, he said, has a one-way return; there is no “re-return.” Now that the district court order has been vacated, he added, there is no underlying decision. The relief provided by the Hague convention has been given.

Cullen explained that a Scottish court would find irrelevant an appellate court’s order that the United States is the child’s habitual residence. The issue before the Scottish court is custody, he said, and that court is focused on where the child has been for the previous 14 months, not for the 18 months before that period.

Still, Chief Justice John Roberts Jr., seemed skeptical. “The incentives if you prevail are for the custodial or the parent with control over the child to leave immediately,” he said. “Even after a motion has been filed—if a motion for a stay has been filed. Get on the first plane out and then you’re home free. That seems to me to be a very unfortunate result.”

Put simply, this case is not moot because an appellate decision on the merits would matter. All the court needs to decide is that the appellate court’s decision would make a difference here.