Immediately afterward, the F.B.I. said that it would spearhead an inquiry into the incident, which they were treating as a “potential assault against a federal officer,” according to Jeanette Harper, a spokeswoman for the agency. Assaults against Customs and Border Protection officers, including Border Patrol agents, reached 720 in the 2017 fiscal year, the most in at least five years, according to data from the agency.

Ms. Harper said the F.B.I. had collected forensic evidence that was being analyzed at its laboratory in Quantico, Va., and was also “aggressively” following up on leads that came into the agency’s tip line.

One of those led to a search warrant being issued for a car belonging to two brothers. But later, Ms. Harper, the F.B.I. spokeswoman, said the agency had not arrested anyone, or even identified anyone “of interest,” in the investigation, suggesting that the tip about the brothers may not have proved fruitful.

Angie Ochoa, who was engaged to marry Mr. Martinez when he died, said she even went to the scene of the incident, escorted by one of her fiancé’s co-workers, after she became fed up with the shifting accounts of what may have happened. “None of it made sense, so I wanted to see for myself,” she said.

Part of the challenge of establishing what exactly happened to the men is the remoteness of their work. Border agents patrol rocky deserts, often in the middle of the night, where there are very few, if any, other people around. The incident in November happened outside a tiny town of less than 2,000 people. The other agent, Mr. Garland, who has been released from the hospital, is said to have no memory of what happened.

Mr. Martinez’s father, Jose Martinez, said in an interview shortly after his son died that after seeing his son in the hospital, he thought the injuries were too severe to have been caused by an accident. “I believe someone put a trap,” he said. The elder Mr. Martinez said he had often worried about his son working alone in the middle of the night.