The Lefschetz condition conjecture for Braden–MacPherson sheaves

Let $w\in{\widehat{\mathcal W}}$ and suppose that $\operatorname{char} K>N(w)$. For every $x\le w$, let

$$i_x\colon\overline{\mathcal B}(w)_x\to\overline{\mathcal B}(w)^x$$

be the indicated map between the costalk and stalk of the reduced Braden–MacPherson sheaf. Recall that a map $f\colon A\to B$ of graded free $K[t]$-modules satisfies the Lefschetz condition with center $l\in\mathbb Z$ if multiplication by $t^n$ induces an isomorphism

$$(\operatorname{coker}f)_{\{l-n\}}\stackrel{\sim}{\to}(\operatorname{coker}f)_{\{l+n\}}$$

for every $n\ge1$. The Lefschetz condition conjecture. For all $x\le w$, the map $i_x$ satisfies the Lefschetz condition with center $l(w)$. This conjecture proposes a Hard Lefschetz-type symmetry for the stalk–costalk maps of Braden–MacPherson sheaves in characteristic greater than $N(w)$. The supplied text does not state whether it has been resolved.