Relative volume of comparable pairs under semigroup majorization

Any semigroup $$\mathcal {S}$$ of stochastic matrices induces a semigroup majorization relation $$\prec ^{\mathcal {S}}$$ on the set $$\Delta _{n-1}$$ of probability n-vectors. Pick X, Y at random in $$\Delta _{n-1}$$ : what is the probability that X and Y are comparable under $$\prec ^{\mathcal {S}}$$ ? We review recent asymptotic ( $$n\rightarrow \infty $$ ) results and conjectures in the case of majorization relation (when $$\mathcal {S}$$ 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 $$\mathcal {S}$$ is the set of upper-triangular stochastic matrices.