A more specific conjecture would be that any optimal dissection (in your extended criterion of tie-breaking) of a set of n m-sized sticks and a set of m n-sized sticks into two identical piles of fragments can be performed by a sequence of "Divide (the remainder of) some stick from one set into N (N>=1) equal sized fragments, and removing the corresponding fragments from sticks in the other set. Then repeat, dividing one of the new remainders, until nothing is left."

I don't think that's worded quite right, but I think something like that. If so, maybe a perturbations argument could show that if at some step you divide into the same number of fragments of slightly different lengths, you can follow through the remainder of the steps in the same way. And if so, the non-equal division would be sub-optimal (because one stick of that length gets shorter).

But I'm not sure if that's actually right, or just another plausible guess that will turn out to have exceptions.

This seems like just the sort of conjecture that we could actually check against the piles of data from the search program! (Which may now be working [crosses fingers], so some data might actually be forthcoming once I've sorted out a way to present it legibly.)