I tried the caviar thing and went through 40 nights with no battles :( I also went through without auto-win on.
Could it be because I maxed out lily's path in the last combat system and so it doesn't count or something?
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Hmm, if you maxed out Lily's path in the old system then you should already have gotten the Lyx event (there's only one).
I'll look into it and see if there's something I didn't notice blocking the event from triggering.
EDIT: Looked into it and immediately found the problem :P It's been fixed for the next update. Sorry for the inconvenience!