Relative volume of comparable pairs under semigroup majorization

Any semigroup S of stochastic matrices induces a semigroup majorization relation ≺S on the set Δn-1 of probability n-vectors. Pick X, Y at random in Δn-1: what is the probability that X and Y are comparable under ≺S? We review recent asymptotic (n→∞) results and conjectures in the case of majorization relation (when S is the set of doubly stochastic matrices), discuss natural generalisations, and prove a new asymptotic result in the case of majorization, and new exact finite-n formulae in the case of UT-majorization relation, i.e. when S is the set of upper-triangular stochastic matrices.