The Ext-algebra conjecture for categories associated with distinguished nilpotents

Let $\mathfrak{g}$ be a Lie algebra, let $e$ be a distinguished nilpotent element, and let $Q=Q(e)$ be the finite centralizer of an associated $sl_2$-triple. Let $\mathcal{S}_e$ be the Slodowy slice at $e$, equipped with its $Q\times\mathbb{C}^{\times}$-action, and let $\mathcal{O}(\mathcal{S}_e)^Q$ be the graded algebra of $Q$-invariant functions. Write ${\bf 1}$ for the tensor unit of $\mathcal{C}(\mathfrak{g},e,l,q)$. The Ext-algebra conjecture. There is an isomorphism of graded algebras

$$\operatorname{Ext}_{\mathcal{C}(\mathfrak{g},e,l,q)}({\bf 1},{\bf 1})\cong \mathcal{O}(\mathcal{S}_e)^Q.$$

The conjecture is motivated by the relation between the cohomology of these categories and the singularity of the nilpotent cone at $e$; the source does not provide a proof or a resolution.