The weak-order chain-length conjecture for alternating sign matrices

Let $A$ be an alternating sign matrix of size $n$, and let $\preceq$ denote the ASM weak order. For each $B\preceq A$, let $X_B$ be the associated variety.

Weak-order chain-length conjecture. The weak-order saturated chains from the identity to $A$ are all of the same length if and only if $X_B$ is Cohen--Macaulay for every $B\preceq A$.

This conjecture proposes an equivalence between a combinatorial property of the interval below $A$ in ASM weak order and Cohen--Macaulayness of all associated varieties in that interval. The supplied source does not indicate whether the claim has been proved or refuted.