Kapranov–Schechtman's total-positivity conjecture for finite Coxeter groups

Let $(W,S)$ be a finite Coxeter group of rank $|S|=n$. For subsets $I,J\subset S$, let $W_I$ and $W_J$ be the parabolic subgroups of $W$ of types $I$ and $J$, respectively. Define the metamatrix $M(W)=(M_{pq})_{0\leq p,q\leq n}$ by

$$M_{pq}=\sum_{I,J\subset S,|I|=p,|J|=q}|W_I\backslash W/W_J|.$$

Kapranov–Schechtman's conjecture. The metamatrix $M(W)$ is totally positive.

This extends the known type-$A$ result, where $W=S_n$, to arbitrary finite real reflection groups. The conjecture is proved in the source for type $B$ and the exceptional types, while the general finite-Coxeter-group case remains open.