Shareshian–Wachs conjecture on torsion in chessboard-complex homology

Let $\mathsf{M}_{m,n}$ denote the chessboard complex, and let $\tilde{H}_{d}(\mathsf{M}_{m,n};\mathbb{Z})$ be its reduced homology. For the parametrization used in the source, let $k$, $a$, and $b$ be the associated integers. Shareshian–Wachs conjecture. For $1\le m\le n$, the group $\tilde{H}_{d}(\mathsf{M}_{m,n};\mathbb{Z})$ contains $3$-torsion if and only if

\left\{\begin{array}{ccl}m\le n\le 2m-5\\\\left\lceil\frac{m+n-4}{3}\right\rceil\le d\le m-3\end{array}\right. \Longleftrightarrow \left\{\begin{array}{ccl}k&\ge&0\\a&\ge&0\\b&\ge&2. \end{array}\right.

This describes precisely the range in which the relevant chessboard-complex homology has $3$-torsion; the supplied context discusses the connection with the $3$-torsion results of Shareshian and Wachs and related homology calculations.