Cohen–Macaulayness of Schubert tangent-cone rings

Let $GL_n/B$ be the complete flag variety, with Schubert varieties $X_w$ indexed by permutations $w\in\mathfrak S_n$. For $v\leq w$ in Bruhat order, let $R_{v,w}$ be the associated graded ring of the local ring of $X_w$ at the torus-fixed point $e_v$, and write its Hilbert numerator as $H_{v,w}(q)$. Cohen–Macaulayness conjecture. The ring $R_{v,w}$ is Cohen–Macaulay. Consequently,

$$H_{v,w}(q)\in\mathbb N[q].$$

This is attributed in the source to Li and Yong and is stronger than Cohen–Macaulayness of the Schubert variety itself; the source notes that it was checked for $n\leq 7$ during preparation of the cited work, but gives no general proof.