Another son, Siamak, 46, a businessman and dual citizen who had been an advocate for improved American-Iranian relations, remains imprisoned as well. Babak said his brother had been abused while in prison, including through electric shock. In a secretive October 2016 trial, the two men were convicted of collaborating with a hostile power — the United States. Their convictions and 10-year sentences were upheld last November on appeal. The details of the charges against them have never been fully explained.

Image Baquer Namazi Credit... Namazi family, via Reuters

At least two other American citizens are known to be imprisoned in Iran: Xiyue Wang, a Princeton graduate student who was seized while working on his doctoral thesis, and Karan Vafadari, a Tehran art gallery owner. Another American, Robert Levinson, has been missing in Iran for more than 10 years. Other foreign citizens have also been detained, including Nazanin Zaghari-Ratcliffe, a Briton of Iranian descent employed by the Thomson Reuters Foundation, who was sentenced to five years.

These detentions have worsened tensions between Iran and the West following the 2015 deal under which Iran agreed to curb its nuclear program in return for a lifting of economic sanctions. The result is that some Westerners, especially those with dual Iranian citizenship, are now more cautious about visiting Iran, fearing arbitrary arrest. Iranian officials say that the United States or its allies have unfairly imprisoned or prosecuted at least 14 Iranians, mostly for what the officials call unfounded charges of sanctions violations.

Granting Mr. Namazi a short-term medical leave and then extending it seemed like one possible way to break the logjam. The Trump administration has reportedly asked Iran to set up a private channel to discuss such humanitarian issues, but there has been no response from Tehran. The speculation is that the hard-liners who control Iran’s judiciary and intelligence services are refusing to budge.

Mr. Trump and his State Department should continue trying to engage Iran. It’s not in the interest of either country for Mr. Namazi, or any other political prisoner, to die in jail.