Mr. Downs’s story has also shone a spotlight on what critics call Qatar’s opaque judicial system, which gave him little opportunity to defend himself during a trial conducted in Arabic, which he does not understand. Neither his family nor American consular officials were permitted to attend most of the proceeding, and his lawyers at the time did little to challenge the prosecution.

His family has since hired another legal team, which contends that Mr. Downs was incorrectly accused of espionage — a capital offense in Qatar — based on an unfounded assumption that he had intended to provide Iran with large quantities of proprietary data from his employer.

“They didn’t know what he was up to — they took every piece of information off his desk and said, ‘This was what you were planning to give Iran,’” said Randy Papetti, one of his lawyers.

Qatari prosecutors asserted that the information was worth billions of dollars. “But Downs never took, let alone offered to sell, all that information as part of his $20,000 proposal to Iran,” Mr. Papetti said.

He described the prosecution of the case and the conviction, which received intense publicity in Qatar at the time, as a mix of misinformation and fact used to reach an absurd outcome.

“You’ve got a nerdy geologist who very wrongly tried to raise some money for his kid’s tuition, when he thought he had been cheated out of his bonus,” the lawyer said. “This was turned into, ‘We caught someone who was a spy.’”

Last month, Mr. Papetti and other lawyers working for Mr. Downs submitted a petition to the United Nations Working Group on Arbitrary Detention, a panel of the Human Rights Council, seeking “urgent action” in their effort to persuade Qatar to release him.