If Puerto Rico’s federal oversight board is declared unconstitutional, what would that mean for the island’s effort to restructure its $74 billion of debt?

That’s the question U.S. District Court Judge Laura Taylor Swain, who is overseeing Puerto Rico’s record bankruptcy, raised Jan. 10 to Aurelius Investment LLC, a bondholder that’s asking the court to strip the current panel of its role in the territory’s financial affairs. It was the first time Swain heard arguments in court by those who are challenging the legitimacy of the board created by the U.S. rescue law enacted in 2016.

Aurelius claims the panel...