Nash blow-up conjecture for covexillary Schubert varieties

Let $X_w^P$ be a Grassmannian Schubert variety. For a covexillary permutation $w$, write

$$\operatorname{Coess}(w)=\{(p_i,q_i)\mid 1\leq i\leq m\},$$

with $p_1\leq\cdots\leq p_m$ and $q_1\leq\cdots\leq q_m$, and let $r_i=r_{p_i,q_i}(w)$. Let $P$ be the largest standard parabolic subgroup such that $w$ is a maximal coset representative for $wW_P$, and let $Z_w$ and $Z'_w$ be the Cortez–Zelevinsky resolutions of $X_w^P$ defined by

$$Z_w=\{(F_\bullet,V_\bullet)\in\operatorname{Fl}(r_1,\ldots,r_m)\times\operatorname{Fl}(q_1,\ldots,q_m)\mid F_{r_i}\subseteq (E_{p_i}\cap V_{q_i})\},$$ $$Z'_w=\{(F_\bullet,V_\bullet)\in\operatorname{Fl}(q_1+p_1-r_1,\ldots,q_m+p_m-r_m)\times\operatorname{Fl}(q_1,\ldots,q_m)\mid F_{q_i+p_i-r_i}\supseteq (E_{p_i}+ V_{q_i})\}.$$

Nash blow-up conjecture for covexillary Schubert varieties. Let $w$ be a covexillary permutation with parabolic $P$ defined as above. The Nash blow-up of $X_w^P$ is isomorphic to

$$Z_w\times_{X_w^P} Z'_w.$$

This conjecture extends the established Grassmannian case, where the Nash blow-up is the fiber product of the two Zelevinsky resolutions. Its status for general covexillary Schubert varieties is not established in the supplied text.