The regularity-depth equality conjecture for chessboard complexes

Let $\Delta_{m,n}$ be a chessboard complex with $n\geq m\geq 1$, and let $S/{\mathcal F}(\Delta_{m,n})$ denote its facet ring. Regularity-depth equality conjecture.

$${\mathbf reg}\,(S/{\mathcal F}(\Delta_{m,n}))=\depth\,(S/{\mathcal F}(\Delta_{m,n}))=2(m-1).$$

The preceding results establish this equality when $m\leq 3$; the conjecture proposes it for all chessboard complexes with $n\geq m\geq 1$.