The freeness conjecture for chessboard-complex homology

Let $M_{m,n}$ be the chessboard complex, let $\nu_{m,n}$ be its bottom nonvanishing homology dimension, and suppose $m\leq n$ and $i\geq\nu_{m,n}$.

Freeness conjecture. The reduced homology group $\widetilde H_i(M_{m,n})$ is free if and only if $i\geq m-2$.

This conjecture extends known freeness results from the diagonal and the case $n=2m-2$ to all homological degrees at or above the bottom nonvanishing degree.