The Kazhdan–Lusztig theorem for finite W-algebras

Let $W_S$ be the Weyl subgroup associated to $\gf_S$, and let $W_S\setminus W$ be the corresponding coset with its Bruhat ordering. For $\sigma\leq\tau$ in this coset, let $L^S_\tau$ denote the irreducible representation and $M^S_\sigma$ the corresponding Verma module, let $\epsilon_\sigma$ and $\epsilon_\tau$ be their signs, and let $\tilde{P}^S_{\sigma,\tau}(x)$ be the dual relative Kazhdan–Lusztig polynomial. Kazhdan–Lusztig theorem. The character of $L^S_\tau$ is

$$\operatorname{ch} L^S_{\tau}=\sum_{\sigma\leq\tau\in W_S\setminus W}\epsilon_\sigma\epsilon_\tau\tilde{P}^S_{\sigma,\tau}(1)\operatorname{ch} M^S_\sigma.$$

This is the proposed character formula for irreducible finite W-algebra representations, with the dual relative Kazhdan–Lusztig polynomials encoding subsingular-vector multiplicities. The supplied text gives no evidence that the theorem has been proved or disproved.