The degree conjecture for Bruhat-graph sheaves

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^y$ be the stalk of the sheaf ${\mathscr B}(x)$ at $y\in{\mathcal W}$. For a graded $S$-module $M$, write $M_{\{\leqslant k\}}$ for the submodule generated in degrees at most $k$. The degree conjecture. For $x,y\in{\mathcal W}$ with $y<x$, the module ${\mathscr B}(x)^y$ is generated in degrees $<l(x)-l(y)$, equivalently,

$${\mathscr B}(x)^y={\mathscr B}(x)^y_{\{\leqslant l(x)-l(y)-1\}}.$$

This conjecture is part of the combinatorial approach to the Kazhdan–Lusztig conjecture and is presented as a degree-generation property of the sheaf stalks. The supplied text gives no resolution status.