Sorry, I should have been more clear. The thing that I thought might help you (to prove the thing about independence of Bayes score) was the discussion of equalizer rules in the paper. I do not know if it will actually be helpful, just pattern-matching between your question and the paper. Incidentally, my intuition is that you need some assumptions (maybe some sort of convexity?) for the result to hold in general.
Sorry, I should have been more clear. The thing that I thought might help you (to prove the thing about independence of Bayes score) was the discussion of equalizer rules in the paper. I do not know if it will actually be helpful, just pattern-matching between your question and the paper. Incidentally, my intuition is that you need some assumptions (maybe some sort of convexity?) for the result to hold in general.