Cohen–Macaulayness characterized by weak-order chains

Prove that, for every alternating sign matrix A, all saturated chains from the identity permutation to A in the ASM weak order have the same length if and only if every ASM variety X_B with B weakly below A is Cohen–Macaulay.

Background

The paper establishes that the weak-order saturated chains from the identity to an alternating sign matrix A are all of the same length if and only if the ASM varieties X_B are equidimensional for every B weakly below A. The authors propose strengthening equidimensionality to the Cohen–Macaulay property, which would connect a combinatorial condition on ASM weak order with a fundamental local-algebraic property of the associated varieties.

The reverse implication is already known because Cohen–Macaulay varieties are equidimensional. The authors further explain that it would suffice to prove that equal-length weak-order chains to A imply that X_A itself is Cohen–Macaulay; they report verification of the conjecture for all ASMs of size n at most 6.