Connectedness and equidimensionality conjecture for enhanced Springer fibers

Let $(X,w)\in\underline{\mathcal{N}}$ be an enhanced nilpotent element of type $\lambda[q]$, and let $(Y,u)\in\overline{\mathcal{O}_{(X,w)}}$. The fiber

$${{}^{\textsf{e}}\hskip-2pt\pi}_{\hskip-2pt\lambda[q]}^{-1}(Y,u)$$

is the enhanced Springer fiber over $(Y,u)$ associated with the type $\lambda[q]$.

Connectedness and equidimensionality conjecture. The fiber ${{}^{\textsf{e}}\hskip-2pt\pi}_{\hskip-2pt\lambda[q]}^{-1}(Y,u)$ is connected, and all of its irreducible components have the same dimension.

This proposes an analogue of Spaltenstein's theorem for Springer fibers in the enhanced nilpotent setting. The source presents it as a proposal related to the Achar–Henderson–Jones conjecture that closures of enhanced nilpotent orbits are normal varieties; its resolution is not specified here.