Finite models for the fixed-point components of Springer fibers

Let $(e,h,f)$ be an $\mathfrak{sl}_2$ triple in $\mathfrak{g}$, let $T_e$ be the torus with Lie algebra $\mathbb{C}h$, and let $\operatorname{Spr}$ be the Springer fiber. Set $\operatorname{Fix}:=\operatorname{Spr}^{T_e}$. The group $A_e$ acts on $\operatorname{Fix}$ and permutes its connected components; write $(\operatorname{Fix})_\alpha$ for the variety corresponding to an $A_e$-orbit of components, and let $Y_e$ be an $A_e$-finite model of $\operatorname{Spr}$. Finite-model conjecture. Each variety $(\operatorname{Fix})_\alpha$ admits a unique $A_e$-finite model $Y_\alpha$. Furthermore, $Y_e$ is the disjoint union of all $Y_\alpha$. This predicts that the fixed-point components have canonical finite models whose union recovers the finite model of the Springer fiber.