The Slodowy-slice identification for type A Hamiltonian reductions

Let $e_1,e_2\in\mathfrak{g}$ be nilpotent elements, and let $\mathfrak{m}_1,\mathfrak{m}_2\subseteq\mathfrak{g}$ be the subalgebras constructed for a covering pair of nilpotent orbits. Write $M_2$ for the algebraic group associated with $\mathfrak{m}_2$, let $\chi_2$ be the corresponding character, and let $\mathfrak{g}^*\mathbin{\sslash_{\chi_2}}M_2$ denote the Hamiltonian reduction. Slodowy-slice identification conjecture. The reduced space

$$\mathfrak{g}^*\mathbin{\sslash_{\chi_2}}M_2$$

is isomorphic to the Slodowy slice $\mathcal{S}_{\chi_2}$ as a Poisson variety. This identification is the classical geometric statement underlying the proposed reduction-by-stages construction for nilpotent orbits; the supplied text does not establish its resolution status.