Regularity equals the degree of the Hilbert numerator for Schubert varieties

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 $e_v=vB/B$ be the corresponding torus-fixed point, let $R_{v,w}$ be the associated graded ring of the local ring of $X_w$ at $e_v$, and write its Poincaré series as

$$\sum_{i=0}^{\infty}\dim(\mathfrak m_{e_v}^i/\mathfrak m_{e_v}^{i+1})q^i=\frac{H_{v,w}(q)}{(1-q)^{\ell(w)}}.$$

Regularity conjecture. The Castelnuovo–Mumford regularity satisfies

$$\operatorname{Reg}(R_{v,w})=\deg H_{v,w}(q).$$

This predicts that the regularity of the tangent-cone ring at a torus-fixed point is determined by the degree of its Hilbert numerator. The source provides no resolution status or supporting evidence beyond stating the conjecture.