The rank-one centralizer conjecture for admissible pairs and finite W-algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra, let $(e,h,f)$ be an $\mathfrak{sl}_2$-triple, and let $\mathfrak{g}^{\mathfrak{s}}$ be the centralizer in $\mathfrak{g}$ of the subalgebra $\mathfrak{s}$ generated by this triple. An $e$-admissible pair consists of the data defined in the paper for a grading associated with $e$. Assume that

$$\operatorname{rk}\mathfrak{g}^{\mathfrak{s}}=1.$$

Rank-one centralizer conjecture. The $e$-admissible pairs are equivalent to one another. In particular, the associated finite $W$-algebras are isomorphic.

The paper proves this assertion when $\mathfrak{g}$ is of classical type or of exceptional type $\mathbf{G}_2$, $\mathbf{F}_4$, or $\mathbf{E}_6$. It remains open in the general rank-one case, and resolving it would be a first step toward the paper's more general conjecture that all $e$-admissible pairs are equivalent.