The canonical-quotient conjecture for Springer-fiber components

Let $e\boxtimes\fg$ be a special nilpotent element. Let $A_e$ be its component group, let $\bar{A}(e)$ be Lusztig's canonical quotient, and let $K_e$ be the kernel of the quotient map

$$A_e\twoheadrightarrow \bar{A}(e).$$

Let $c_e$ be the corresponding two-sided cell, let $H_\sigma\subset \bar{A}(e)$ be the subgroup associated with each left cell $\sigma$ in $c_e$, and set

$$Y=\bigsqcup_{\sigma}(\bar{A}(e)/H_\sigma).$$

Canonical-quotient conjecture. The $\bar{A}(e)$-sets $\Irr(\spr)/K_e$ and $Y$ are isomorphic.