The phrase "apocryphal story" refers to a tale or narrative that may not be true or have enough evidence to support it. It's like a story that might be made up or exaggerated, with doubts about its accuracy. Full definition
There's a possibly apocryphal story of Lord Denning MR changing his mind on who won an appeal after the judgment had been released to the WLR so that the WLR version has the plaintiff winning 2 - 1 while the All ER version had the defendant winning 2 - 1. (slaw.ca)
There is a possibly apocryphal story about one of the early twentieth century editors, Ring Lardner I think, memory is failing me now. (madgeniusclub.com)
A possibly apocryphal story has Thoreau standing by his pond and watching the first telegraph lines from Maine to Boston being strung and saying, «What if they have nothing to say to each other?» (religion-online.org)