Cohen–Macaulayness conjecture for the ASM embedding $A\mapsto 1\oplus A$

Let $A\in\operatorname{ASM}(n)$ be an alternating sign matrix, and let $1\oplus A\in\operatorname{ASM}(n+1)$ be the direct sum of the $1\times1$ identity matrix with $A$. Cohen–Macaulayness conjecture. The ASM $A$ is Cohen–Macaulay if and only if $1\oplus A$ is Cohen–Macaulay. The conjecture is motivated by the proved equidimensionality equivalence and by computer calculations over the rational numbers for $n\leq6$. The paper also proves that Cohen–Macaulayness of $1\oplus A$ implies Cohen–Macaulayness of $A$, leaving the converse as the substantive open direction.