The Franklin Institute, a Philadelphia museum dedicated to the study of science, long has said one of its most impressive holdings — a plane built in 1911 by the Wright brothers — was a gift from a man named Grover C. Bergdoll.

The Wright Model B, a two-seater that Bergdoll bought from the brothers, remains one of the best-preserved icons of early aviation. The museum’s website details the plane’s rich history and how it was built with inventive flaps and cables.

But it says nothing about the man the museum says was the plane’s donor, a wealthy bon vivant who was utterly despised after dodging the draft for World War I. The scion of a Philadelphia brewing fortune, Bergdoll drove cars and flew planes with an abandon before the war that earned him the nickname “Playboy of the Eastern Seaboard.”

For nearly a century, that plane has been exhibited at the Franklin Institute. But more recently, the circumstances of how it got there have become a point of contention.

The Franklin Institute acquired the plane in 1933, when Bergdoll was living as a fugitive in Germany, to which he had fled after his conviction for desertion. By this time, all of his possessions had been declared the property of the U.S. government. The museum has said in several settings that Bergdoll transferred title by letter while he hid from U.S. authorities overseas.

But recently the museum acknowledged, to the author of a book on Bergdoll and Bergdoll’s family, that it has no letter. Instead, a museum official said, Bergdoll had simply told a museum official verbally that he wanted to give the plane to the museum.

“From your own knowledge of Mr. Bergdoll and his background,” a museum curator, Susannah Carroll, wrote to the author, Timothy W. Lake, “you should understand why neither he nor The Institute would desire to have anything in writing documenting the oral gift. Bergdoll was still a fugitive and his assets had been and continued to be subject to government seizure.”

In recent months, though, Bergdoll’s family has challenged the museum’s account. In an interview, one of Bergdoll’s daughters, Katharina, described the institute’s explanation as inconsistent and said it does not address the fact that the government had placed all her father’s assets under seizure more than a decade earlier.

“Getting a verbal agreement — how was it possible when my father was a fugitive at the time in Germany?” she said. “You could not have reached him. That was the first impossibility. The second: It was technically in the government’s possession at the time. He could not have legally transferred it.”

Family members have now asked the museum to consider returning the plane, or to agree to some other form of compensation. Katharina Bergdoll said she would also like the museum to “own up to the facts of how it was obtained.”

The Bergdoll family said that, while it believes in the merits of its claim, it is not trying to rehabilitate the reputation of a man who became a figure of national loathing and in the end paid for his behavior.

Although there were other so-called slackers, Bergdoll’s draft-dodging drew particular scorn because of his wealth, high profile and demonstrated disinterest in following the rules.

By 1917, when he failed to report for military service, Bergdoll had already taken flying lessons from Orville Wright, had bought the plane for $5,000 and had made a habit of buzzing buildings and other stunts.

His reckless driving, multiple accidents and traffic violations as a teenager earned him the moniker “speed fiend” in the local press, and he served two months in jail after a head-on crash in 1913.

After he disappeared rather than serve in World War I, the government made an example of him, distributing his image on wanted posters across the country. When the man thought to have been drafted because Bergdoll didn’t show was shot to death in a French forest, The New York Times carried news of his death on Page 1: “Died Hero in Battle in Bergdoll’s Place.”

Bergdoll eventually was caught in 1920 when a nationwide manhunt ended with his being found, hiding inside a window seat at his family’s gray stone mansion in Philadelphia. Sentenced to five years in prison, he served just a few weeks before managing a daring escape that transfixed the nation.

Though museum procurement procedures at the time could be haphazard, Lake, a former Philadelphia television news anchor, said he found it odd that an institution as celebrated as the Franklin would have relied simply on a donor’s word.