Soergel's conjecture on the character of indecomposable Soergel bimodules

Let $(W,S)$ be the Coxeter system underlying the Soergel bimodule category, let $B_x$ denote the indecomposable Soergel bimodule indexed by $x\in W$, and let $b_x$ be the corresponding Kazhdan–Lusztig basis element of the Hecke algebra $H$. The character map is denoted by $ch$. Soergel's conjecture. For all $x\in W$,

$$ch(B_x)=b_x.$$

This conjecture was proved by Elias and Williamson when the coefficient field $\mathbbm{k}$ has characteristic $0$. In characteristic $p>0$, the assertion fails; the characters of the indecomposable bimodules instead give the $p$-canonical basis.