Coefficientwise semicontinuity of Schubert Hilbert numerators

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 $H_{u,w}(q)$ and $H_{v,w}(q)$ be the Hilbert numerators of the associated graded local rings at $e_u$ and $e_v$, respectively, and write $[q^t]H(q)$ for the coefficient of $q^t$ in $H(q)$. Coefficientwise semicontinuity conjecture. If $u\leq v\leq w$, then

$$[q^t]H_{u,w}\geq[q^t]H_{v,w}.$$

The source describes this as a strengthening of the Cohen–Macaulayness conjecture and notes that it was checked for at least $n\leq 6$ and much of $n=7$. No general proof or refutation is given.