Fiebig's conjecture on moment-graph sheaves and the Kazhdan-Lusztig basis

Let $\widehat{W}^{\circ}$ be the finite subset of an affine Weyl group used in the moment-graph construction, let $\mathcal{B}(w)$ be the indecomposable object associated with $w\in\widehat{W}^{\circ}$, and let $\underline{\mathrm{rk}}\mathcal{B}(w)^x$ denote the graded rank of its $x$-component. Let $C'_w$ be the Kazhdan–Lusztig basis element and $T_x$ the standard Hecke-algebra basis element. Fiebig's conjecture.

$$C'_w=\sum_{x}\underline{\mathrm{rk}}\mathcal{B}(w)^xT_x\quad\text{for all }w\in\widehat{W}^{\circ}.$$

The paper states that this conjecture implies Lusztig's conjecture for the dual group and later proves that it is equivalent to affine Soergel's $p$-conjecture. Its resolution is therefore tied to the corresponding positive-characteristic Soergel-bimodule statement.