Braden–MacPherson stalk-rank conjecture

Let $R$ be a reduced, irreducible, finite root system, let $\widehat{\mathcal W}$ be its affine Weyl group, and let $k$ be a field. Form the associated moment graph $\widehat{\mathcal G}_k$ over $W_k=\widehat X^\vee\otimes_{\mathbb Z}k$. For $w\in\widehat{\mathcal W}$, let $\mathcal B_{k,w}$ be the space of global sections of the Braden--MacPherson sheaf on the truncated moment graph, and let $\mathcal B_{k,w}^x$ be its stalk at $x$. Define the graded rank by

$$\underline{\operatorname{rk}}\,\mathcal B_{k,w}^x=\sum_i v^{2l_i}$$

when $\mathcal B_{k,w}^x\cong\bigoplus_i S\{l_i\}$. Let $P_{x,w}$ be the Kazhdan--Lusztig polynomial. Braden–MacPherson stalk-rank conjecture. If $k$ and $w$ are such that $\widehat{\mathcal G}_{k,w}$ is a GKM-graph, then

$$\underline{\operatorname{rk}}\,\mathcal B_{k,w}^x=P_{x,w}(v^{-2})$$

for all $x\leq w$. This conjecture is the moment-graph multiplicity conjecture underlying the paper's approach to Lusztig's conjecture; the supplied text does not state that it has been resolved.