Cohen–Macaulay tangent-cone conjecture for Kazhdan–Lusztig varieties

Let $R={\mathbb C}[{\bf z}^{(v)}]$, let $I_{v,w}$ be the Kazhdan–Lusztig ideal defining the local chart of $X_w$ at $E^{(v)}_\bullet$, and let $\overline I_{v,w}$ be its ideal of lowest-degree terms. Write $H_{v,w}(t)$ for the numerator of the Hilbert series of $R/\overline I_{v,w}$. Tangent-cone Cohen–Macaulay conjecture. The quotient $R/\overline I_{v,w}$ is Cohen–Macaulay. Hence $H_{v,w}(t)\in{\mathbb N}[t]$. A positive answer would give strong control of tangent cones and their Hilbert series; the supplied text does not state a resolution status.