The GKM-pair character conjecture for Braden–MacPherson sheaves

Let ${\widehat{\mathcal G}}$ be the affine moment graph, let $w\in\mathcal V={\widehat{\mathcal W}}$, and let $(K,w)$ be a GKM-pair. For the Braden–MacPherson sheaf ${\mathscr B}(w)$, define

$$h({\mathscr B}(w)):=\sum_{x\le w}{{\operatorname{rk}}}^\prime\,{\mathscr B}(w)^x v^{l(w)}T_x,$$

where, for a graded free $S$-module $M\cong\bigoplus_{i=1}^n S\{l_i\}$, ${{\operatorname{rk}}}^\prime M:=v^{l_1}+\dots+v^{l_n}$. Let $\underline{H}_w$ denote the self-dual affine Kazhdan–Lusztig basis element of the affine Hecke algebra associated with $w$. The GKM-pair character conjecture. If $(K,w)$ is a GKM-pair, then

$$h({\mathscr B}(w))=\underline{H}_w.$$

This conjecture predicts that the graded character of the Braden–MacPherson sheaf agrees with the corresponding affine Kazhdan–Lusztig basis element whenever the pair satisfies the GKM condition. It links moment-graph sheaves with affine Hecke-algebra combinatorics; the supplied text does not state whether it has been resolved.