The Virasoro-type reduction conjecture for adjacent nilpotent orbits

Let $\mathbb{O}$ be a nilpotent orbit of $\mathfrak{g}$, with representative $f\in\mathfrak{g}$ satisfying

$$\dim(\mathfrak{g}^f\cap\mathfrak{h})=1,\qquad \dim(\mathfrak{g}^f\cap\mathfrak{n}_+)=1,$$

and assume that the nonzero semisimple element of $\mathfrak{g}^f\cap\mathfrak{h}$ acts nontrivially on $\mathfrak{g}^f\cap\mathfrak{n}_+$. Suppose also that the corresponding generator $G^+$ satisfies $G^+(z)G^+(w)\sim0$, so that the Virasoro-type BRST reduction $\mathrm{H}^0_{\ydiagram{2}}$ is defined. The Virasoro-type reduction conjecture. One has

$$\mathrm{H}^0_{\ydiagram{2}}(\mathcal{W}^\mathsf{k}(\mathfrak{g},\mathbb{O}))\simeq\mathcal{W}^\mathsf{k}(\mathfrak{g},\widehat{\mathbb{O}}),$$

where $\widehat{\mathbb{O}}$ is a nilpotent orbit adjacent to $\mathbb{O}$ in the Hasse diagram, namely the smallest nilpotent orbit whose boundary contains $\mathbb{O}$. This predicts that the smallest BRST reduction relates W-algebras attached to adjacent nilpotent orbits; it is presented as an expectation and remains open in general.