The principal nilpotent-pair conjecture for non-exceptional pairs

Let $\mathfrak g$ be a simple Lie algebra with Cartan subalgebra $\mathfrak h$, let $\mathfrak g^d=\mathfrak g\oplus\mathfrak g$, and let $\mathbf e=(e_1,e_2)\in\mathfrak g^d$ be a principal nilpotent pair. Let $\mathcal G^{\mathbf e}$ be the associated subgroup of finite index in the automorphism group of $\mathfrak g$, let $\mathcal O_{\mathbf e}$ be its diagonal orbit, and let $W$ be the Weyl group. Define

$$\mathfrak D_{\mathbf e}=\mathbb C[\overline{\mathcal O}_{\mathbf e}\cap\mathfrak h^d],$$

where the intersection is scheme-theoretic. Assume that $\mathbf e$ is non-exceptional. Principal nilpotent-pair conjecture. The scheme $\overline{\mathcal O}_{\mathbf e}\cap\mathfrak h^d$ is Gorenstein, and $\mathfrak D_{\mathbf e}$ supports the regular representation of $W$; in particular,

$$\dim_{\mathbb C}\mathfrak D_{\mathbf e}=|W|.$$

This conjecture is presented as a variant of a conjecture of Ginzburg; the original version is false, while the stated non-exceptional variant is the main conjecture of the paper.