Vexillary partial permutation minimality conjecture for real matrix Schubert varieties

Let $\omega$ be a partial permutation, and let $X_\omega\subset\mathfrak{M}_{m,n}(\mathbb{R})$ denote the corresponding open dense regular part of the real matrix Schubert variety. A partial permutation is vexillary if it avoids the $2143$-pattern. Vexillary partial permutation minimality conjecture. If $\omega$ is a vexillary partial permutation, then $X_\omega\subset\mathfrak{M}_{m,n}(\mathbb{R})$ is minimal. The theorem preceding this conjecture proves the necessary direction for non-vexillary partial permutations; the conjecture asserts that vexillarity is also sufficient for minimality, linking pattern avoidance with the mean-curvature geometry of matrix Schubert varieties.