The Kac determinant formula for finite W-algebras

Consider a finite W-algebra associated to an ${\bf sl}_2$ embedding that is principal in $\gf_S\subseteq\gf$. Let $M(\Lambda)$ be a Verma module, let $\bar{\beta}$ be a position in its weight lattice, let $\bar{\Delta}_+^S$ denote the positive restricted roots, and let $P(\bar{\gamma})$ be the Kostant partition function for restricted roots with their multiplicities. The W-weights are parametrised by a $\gf$-weight $\Lambda$ through invariants of the Weyl group $W_S$ of $\gf_S$. Kac determinant formula. The Kac determinant at position $\bar{\beta}$ is

$$M_{\bar{\beta}}(\Lambda)=\prod_{k>0}\prod_{\alpha\in\bar{\Delta}_+^S}\left(\langle\Lambda+\rho,\alpha\rangle-\frac{k}{2}\langle\alpha,\alpha\rangle\right)^{P(\bar{\beta}-k\bar{\alpha})}.$$

This formula describes the reducibility loci of finite W-algebra Verma modules and underlies the classification of completely degenerate representations. The source presents it as a conjectural formula, but gives no resolution status.