Equidimensionality conjecture for polarized diagonal initial ideals

Let $I_w$ be the relevant matrix Schubert ideal, let $\prec'$ be the lex-diagonal term order, and let $\widetilde{J}$ be the ideal obtained by polarizing $\operatorname{init}_{\prec'} I_w$. The associated Stanley–Reisner complex is denoted by $\Delta_{\widetilde{J}}$. Equidimensionality conjecture. Under $\prec'$, the ideal $\widetilde{J}$ is equidimensional. This conjecture is verified for $n\leq 7$ when $\prec'$ is the lex-diagonal order. It motivates questions about whether $\Delta_{\widetilde{J}}$ is Cohen–Macaulay, shellable, vertex-decomposable, or a subword complex; these properties are known in the antidiagonal case and for diagonal term orders under the vexillary hypothesis.