The birational-sheet degree conjecture for Sommers-Achar stabilizers

Let $\mathcal{O}$ be the special nilpotent orbit occurring in the conformal limit, and let $C$ be the Sommers-Achar subgroup associated with one of the Hitchin images corresponding to $\mathcal{O}$. Let $A(\mathcal{O})$ be the component group of $\mathcal{O}$, and let $Z_C$ denote the stabilizer of $C$ as a subgroup of $A(\mathcal{O})$. Let $\mu$ be the generalized Springer map associated with the sheet corresponding to the mass-deformed defect.

Birational-sheet degree conjecture. The degree of $\mu$ equals the order of the stabilizer:

$$\operatorname{deg}(\mu)=\lvert Z_C\rvert.$$

This conjecture proposes a compatibility between Losev's theory of birational sheets and the correspondence between Hitchin images, Sommers-Achar subgroups, and mass deformations. It relates the degree of a generalized Springer map to the associated stabilizer size.