The positivity conjecture for local Braden–MacPherson sections

Let $\mathcal W$ be the Coxeter group with Bruhat order $<$ and length function $l$. For $x,y\in\mathcal W$ with $y<x$, let $B(x)^{[y]}$ denote the corresponding graded $S$-module associated with the indecomposable projective object $B(x)$. Positivity conjecture. The $S$-module $B(x)^{[y]}$ lives in degrees

$$> -l(y).$$

One of the main problems in Kazhdan--Lusztig theory is to establish this strict lower-degree bound, which is presented in the source as a conjecture; no resolution status is supplied.