As far as I know none of the three are mutually exclusive.

That said, a Lich quest is a custom quest run by an immortal that involves a non-trivial amount of time be invested, so it's probably less likely you'll be taken seriously for that if you've already started and bailed on a Mummy quest. We don't like sloppy seconds.

And of course as has been said before, nobody should ever go into playing a necro with the expectation that they'll be obliged a Lich quest. That's why two other fully automated routes to undeaddom exist, one being a never-fail proposition.