Soergel's character conjecture for indecomposable projectives

Let $\mathcal W$ be the Coxeter group, let $\mathcal H$ be its Hecke algebra, and let $B(x)$ be the indecomposable projective object associated with $x\in\mathcal W$. For a graded $S$-module $N\cong\bigoplus_i S\{k_i\}$, define its graded dimension by

$$\operatorname{\underline{\dim}}(N)=\sum_i v^{-k_i}.$$

For $M\in\mathcal V$, define its graded character by

$$h(M)=\sum_{y\in\mathcal W}\operatorname{\underline{\dim}}(M^{[y]})v^{l(y)}\widetilde T_y\in\mathcal H.$$

Soergel's character conjecture. For any $x\in\mathcal W$,

$$h(B(x))=C_x^\prime.$$

This asserts that the graded character of each indecomposable projective is the corresponding Kazhdan--Lusztig self-dual element. The source attributes the statement to Soergel and gives no resolution status.