Good guess, but here are (additional) counter-examples:
Yes, I just require that all chains have at least one liberty.
With passes allowed, I believe all such positions can be reached in a normal go game, even without captures. With passes not allowed, that could be an interesting question. But a question for another thread
Hypothesis: White is always in the minority when touching black chains.
More precisely: if you count the stones in a white chain and the stones in all black chains touching this white chain, then either you have counted more black stones than white stones, or the white chain is not touching any black stones at all.