Quantum Drinfeld–Sokolov reduction equivalence for generic parameters

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra, let ${}^L\mathfrak{g}$ be its Langlands dual, and let $\widehat{{}^L\mathfrak{g}}_{{}^Lk}\text{-mod}^0$ be the semisimple category of affine Kac–Moody modules described in the source. Let ${\mc W}_\beta(\mathfrak{g})\text{-mod}^0$ be the category whose irreducible subquotients are the modules $H_{\operatorname{DS}}(L_{\lambda,{}^Lk})$, for $\lambda\in{}^LP^+$, where $H_{\operatorname{DS}}$ is quantum Drinfeld–Sokolov reduction and $\beta,{}^Lk$ satisfy equation. Quantum Drinfeld–Sokolov reduction conjecture. If $\beta$ and ${}^Lk$ are generic parameters satisfying equation, then $H_{\operatorname{DS}}$ establishes an equivalence of abelian categories

$$\widehat{{}^L\mathfrak{g}}_{{}^Lk}\text{-mod}^0\simeq {\mc W}_\beta(\mathfrak{g})\text{-mod}^0.$$

The source explains that the affine category is semisimple for generic ${}^Lk$, so the assertion is equivalent to the absence of nontrivial extensions among the reduced modules. The conjecture is not resolved in the supplied text.