semi-silly question: can you precrown with 3 different deities?

There's sort of two parts to this question. I know that you can precrown and then change alignment. I've done that, with one shift. If I remember correctly, (a) you have to be a little careful when you start sacrifices to the 2nd deity, and (b) the first deity you precrowned with gets pretty pissed. I've been thinking about how expensive 2nd/3rd/4th precrowns are to earn.

What I'm picturing here is probably easiest for a character who plans to be C. Precrown L, kill a beggar, precrown N, kill a beggar, precrown C.

As long as this character planned to stay chaotic, is there any substantive problem with that idea? My understanding is that the L and N deities will remain pissed at the character, but as long as the char stays C, that shouldn't impact anything.

My main question, though, is this: Will the N precrown require the amount of sacrifices for a 1st precrown or for a 2nd? Will the C precrown require the amount of sacrifices for a 1st precrown or a 3rd? If the increase in piety is deity-specific, then this would be a way for (a real jerk of) a character to get 3 random artifacts at lower cost.

Changing alignment has no effect on the increased costs from praying for the same thing multiple times.

That makes sense. I suppose the only advantage would be the ability to sacrifice the artifact from the first god to the 2nd god, then. Again, for a real jerk of a character.