Lovecraft typed up the story, but in the most rom-com-esque event in his life, lost it at a train station before leaving for his wedding. So during his honeymoon, he had to retype the story with the help of his new wife. Houdini ended up liking this version of the account, now clearly and absurdly fictional, just fine. Evidently, he never really cared about fooling people, and just wanted a fantastical story. The story was in first-person and was credited solely to Houdini, until a reprint mentioned Lovecraft in an editor's note after both men had died.
Years after writing "Imprisoned With The Pharaohs," Lovecraft needed work again and visited Houdini's Manhattan apartment. (Fun exercise: Imagine Houdini loudly challenging Lovecraft to an arm wrestling match, and then losing.) Later still, Houdini commissioned Lovecraft to write an article debunking astrology, and after this he commissioned him again, this time to ghostwrite a full book on debunking called The Cancer Of Superstition. Houdini soon died, cutting this project short, and the manuscript was thought to be lost until it turned up in 2016. But somehow we don't yet have a Netflix series about the two of them traveling the world, debunking ghosts and escaping elaborate traps.