not necessarily incompatible in all senses.

so is this only a concern/possibility when the word in question is a polyseme?

but there is nothing preventing the queen from being a mother

The meaning of the "incompatibility" relation seems really ambiguous. What does that mean precisely?

And how would we know for sure if an incompatibility (such as a peach is not a plum) or lack of incompatibility (a queen can be a mother and a mother can be a queen) is a sufficient condition to cause it to be or not be a co-hyponym?

Oh. I guess it says

Co-hyponyms are often but not always related to one another by the relation of incompatibility.

so it actually can't ever be used to prove or disprove (sufficient/necessary condition) that something is a co-hyponym. So that observation, while interesting, is not helpful in a practical / deterministic way...

Is this a polyseme?

Or is that only the case if the different distinct senses are all within the same "field"?