Probably not but I think its Jack Napier because its the one that's most memorable.

There is also precedent for it. First in the Burton movie, then he called himself that in White Knight. Then it was his real name in Tales from the Dark Multiverse: Batman Hush

Also in Batman the Animated Series

Those are still alternate worlds. And seeing that main universe doesnt even have just one single Joker, its not really applicable. Not that it would matter as many characters have different names and identities in other worlds. So as it stands, he has no name known to the fans (although Batman knows)

There's only one Joker now, but even though it is confirmed that The Killing Joke's origin is the real origin, we still have no clue about his name.

Thats because the other two are dead lol

Although ive made the argument that only that Joker who survived is the real Joker who we have gone in depth with in the past and the other 2 just arent Joker, thats not really relevent here.

Isn't Jack White one half of the White Stripes?

Joker's name? Yeah, it's still unknown. Crazy, huh? It even says that on his page.

He doesn’t have a name,nor should he.