Babson–Reiner connectivity conjecture for Coxeter-like complexes

Let $T$ be a tree with $n$ nodes and $b$ leaves, and let $cDelta_T$ be its Coxeter-like complex. Babson–Reiner conjecture. The complex $cDelta_T$ is at least $(n-1-b)$-connected. The paper states that this conjecture is proved in its Theorem, so it is solved; it generalizes connectivity results for chessboard complexes and type A Coxeter complexes.