The genericity conjecture for Bruhat-graph sheaf inclusions

Let $({\mathcal W},{\mathcal S})$ be a Coxeter system with length function $l$, and let ${\mathscr B}(x)^{[y]}\hookrightarrow {\mathscr B}(x)^y$ be the inclusion of the costalk into the stalk for $y<x$. For a graded module $M$, let $M_{\{\leqslant k\}}$ denote the submodule generated in degrees at most $k$. The genericity conjecture. For every $x,y\in{\mathcal W}$ with $y<x$, this inclusion is generic in the sense that, for every $m\geq 1$,

$${\mathscr B}(x)^{[y]}_{\{\leqslant l(x)-l(y)+m-1\}}\cap {\mathscr B}(x)^y_{\{\leqslant l(x)-l(y)-m\}}=0.$$

The paper proves that this conjecture implies the degree conjecture, placing it in the proposed inductive route to the Kazhdan–Lusztig conjecture. Its resolution status is not specified in the supplied text.