One of the biggest police misconduct cases in recent New York history, a criminal proceeding that riveted the city this winter and was swept up in the national debate about race and law enforcement, was threatened on Wednesday by an unusual and unexpected question: Did one of the jurors, a retired carpenter, hide his father’s criminal past in order to be a panelist at the trial?

From the start, the case of Peter Liang, a former Brooklyn police officer convicted in February of manslaughter for the shooting of Akai Gurley in a housing project stairwell, has seen its share of twists and turns. Though the three-week trial was entangled in the furor over the death of Eric Garner on Staten Island, it never neatly fit the narrative of other police killings of unarmed black men around the country.

There was, for instance, no direct confrontation between Mr. Liang and Mr. Gurley, who died from a ricocheting bullet that Mr. Liang fired while on patrol in November 2014 in a dark stairwell of the Louis H. Pink Houses in East New York. And even the man who pursued the case, Ken Thompson, the Brooklyn district attorney, recommended after the trial that Mr. Liang should not face time in prison, acknowledging that while he acted recklessly, there was no real proof that he meant to kill or injure Mr. Gurley.

But the story that unfolded on Wednesday — that of the retired carpenter, Michael Vargas, 62, and whether he had lied about his father — took the case into its strangest place yet: It was suddenly possible that the entire prosecution could be thrown out because of what a juror had said about his private life.