As far as I can tell, mine never ate it. I've tried wilting it before putting it in, as well as just spreading it around the coop fresh to wilt on it's own, and they didn't seem to eat it.
As you said, mine love cabbage. I've only tried the red kind, but the taste is similar
enough that I think they would eat any kind. I'm going to try growing much more of it to feed them this year since it keeps fairly well.