Semicontinuity of regularity along Bruhat order

Let $GL_n/B$ be the complete flag variety, with Schubert varieties $X_w$ indexed by permutations $w\in\mathfrak S_n$. For $u\leq v\leq w$ in Bruhat order, let $R_{u,w}$ and $R_{v,w}$ denote the associated graded rings of the local rings of $X_w$ at the torus-fixed points $e_u$ and $e_v$, respectively. Semicontinuity conjecture. If $u\leq v\leq w$ in Bruhat order, then

$$\operatorname{Reg}(R_{u,w})\geq\operatorname{Reg}(R_{v,w}).$$

This predicts that regularity does not increase when moving upward through the interval in Bruhat order. The source gives no resolution status or evidence beyond the conjectural assertion.