The Kazhdan–Lusztig character conjecture at negative level

Let $\mathfrak g$ be a symmetrizable complex Kac–Moody algebra with Weyl group $\mathcal W$ and simple reflections $\mathcal S$. Fix a Weyl vector $\rho$, let $w.\lambda=w(\lambda+\rho)-\rho$ be the dot action, and let $\leqslant$ be the usual partial order on weights. For a weight $\lambda$, call it non-critical when $2(\lambda+\rho,\delta)\notin\mathbb Z(\delta,\delta)$ for every imaginary root $\delta$, integral when $\langle\lambda,\alpha^\vee\rangle\in\mathbb Z$ for every simple root $\alpha$, regular when $w.\lambda=\lambda$ implies $w=e$, and anti-dominant when $\langle\lambda,\alpha^\vee\rangle\notin\mathbb Z_{\geq 0}$ for every simple root $\alpha$. Let $L(\lambda)$ be the irreducible highest-weight representation and $\Delta(\mu)$ the Verma module of highest weight $\mu$. Kazhdan–Lusztig character conjecture. Suppose $\lambda$ is non-critical, integral, regular and anti-dominant. For every $w\in\mathcal W$,

$$\operatorname{ch} L(w.\lambda)=\sum_{y\leqslant w}(-1)^{l(w)-l(y)}P_{y,w}(1)\operatorname{ch}\Delta(y.\lambda),$$

where $P_{y,w}\in\mathbb Z[v]$ is the Kazhdan–Lusztig polynomial associated with $y$ and $w$ for the Coxeter system $(\mathcal W,\mathcal S)$. This generalizes the affine negative-level conjecture, which was proved by Kazhdan and Lusztig; the stated generalization is resolved in the source context.