The Hard Lefschetz isomorphism conjecture for Bruhat-graph sheaves

Choose a generic line and the associated graded algebra ${\mathbb R}[T]$, with $T$ of degree $2$, and let ${\mathscr B}(x)^{\delta y}_T$ be the corresponding module. The Hard Lefschetz property. For $x,y\in{\mathcal W}$ with $y<x$, multiplication by $T^m$ induces an isomorphism

$$T^{m}:{\mathscr B}(x)^{\delta y}_{T,\{l(x)-l(y)-m\}}\stackrel{\sim}{\longrightarrow}{\mathscr B}(x)^{\delta y}_{T,\{l(x)-l(y)+m\}}$$

for every $m\geq 1$. This is the second Hard Lefschetz formulation in the paper and is presented after the decomposition conjecture; the supplied text does not state whether it has been resolved.