The Hard Lefschetz decomposition conjecture for Bruhat-graph sheaves

Choose a generic line in $V^\ast$ and let $\rho:S\to{\mathbb R}[T]$ be the corresponding graded surjection, with $T$ of degree $2$. For an $S$-module $M$, set $M_T=M\otimes_S{\mathbb R}[T]$. Let ${\mathscr B}(x)^{\delta y}_T$ be the indicated quotient module associated with the stalk and costalk of ${\mathscr B}(x)$ at $y$. The Hard Lefschetz decomposition conjecture. For every $x,y\in{\mathcal W}$ with $y<x$, the ${\mathbb R}[T]$-module ${\mathscr B}(x)^{\delta y}_T$ is isomorphic to a direct sum of modules of the form

$$\left({\mathbb R}[T]/T^{n+1}\right)\{-(l(x)-l(y))+n\},\qquad n\geq 0.$$

This is the algebraic Hard Lefschetz-type statement used to imply the genericity conjecture, and hence the degree conjecture. The supplied text gives no resolution status.