Kac–Roan–Wakimoto's Drinfeld–Sokolov reduction conjecture

Let $\mathfrak{g}$ be a Lie algebra, let $f$ be a nilpotent element of $\mathfrak{g}$, and let $L_k(\mathfrak{g})$ be the simple affine vertex algebra at level $k$. Let $H^0_{DS,f}(-)$ denote Drinfeld–Sokolov reduction with respect to $f$, and let $\mathcal{W}_k(\mathfrak{g},f)$ denote the corresponding simple $\mathcal{W}$-algebra. Kac–Roan–Wakimoto's conjecture.

$$H^0_{DS,f}(L_k(\mathfrak{g})) \text{ is either zero or isomorphic to } \mathcal{W}_k(\mathfrak{g},f).$$

The source says that this has been proved in many cases, mainly when $k$ is admissible, and presents it as unresolved in general.