The type A reduced W-algebra identification conjecture

Let $\mathfrak{g}$ be a Lie algebra, let $\mathfrak{m}_2\subseteq\mathfrak{g}$ be the subalgebra associated with a nilpotent element $e_2$, and let $\chi_2$ be its character. Denote the corresponding quantum Hamiltonian reduction by

$$U(\mathfrak{g})\mathbin{\sslash_{\chi_2}}\mathfrak{m}_2.$$

Reduced W-algebra identification conjecture. The reduced algebra is isomorphic to the W-algebra $U(\mathfrak{g},e_2)$:

$$U(\mathfrak{g})\mathbin{\sslash_{\chi_2}}\mathfrak{m}_2\simeq U(\mathfrak{g},e_2).$$

This is the quantum counterpart of the corresponding classical reduction of Slodowy slices and is intended to identify the intermediate reduction obtained by the two-stage construction; the supplied text gives no evidence resolving its status.