(PS: In case it makes a difference, I explain the reason for the vagueness of the initial question. I used such a bound for a very nice, symmetric collection of matrices in the middle of a calculation in a recent paper, and was able to do everything by hand. I now have a bunch of examples where I'd like to do `the same' calculation, and this is the one step where I seem to be using the symmetry of the initial calculation in a way that I don't understand.)

Dear Steve, fedja: Thank you for the thought and the comments. I added one precise (though perhaps overly optimistic) conjecture, and related it to mixing times. As I say, I am also interested in partial results, but I understand that it is a little unfair to ask for partial results without saying what they are part of! Thanks again.