Cohen–Macaulayness under block-sum stabilization

Determine whether an ASM variety X_A is Cohen–Macaulay if and only if the ASM variety X_{[1]\oplus A}, obtained by adjoining a 1-by-1 identity block to A via block sum, is Cohen–Macaulay.

Background

The paper notes that very little is known about Cohen–Macaulayness of ASM varieties. In particular, the authors identify the preservation of Cohen–Macaulayness under the block-sum operation [1]\oplus A as an unresolved question.

The proposed equivalence was conjectured in the cited work \cite{AFH+}. The paper observes that this equivalence would follow from the preceding conjecture characterizing Cohen–Macaulayness by equal-length weak-order chains, making it a concrete consequence of resolving that broader conjecture.