Just looked up the etymology of prestige and it’s interesting.

It comes from Latin praestigium ("delusion, illusion"), then 1500s French prestige meaning “deceit, imposture, illusion”. In the 1800s it started to mean “an illusion as to one's personal merit or importance, a flattering illusion”.

I would have wrongly guessed it originally meant “good reputation” (same as the article author meant it, I assume) and that the association with bullshit/fakery is just a modern twist from people using the word with cynical irony. But bullshit/fakery was in fact the core meaning.