Completeness of component orbits in fixed-point Springer fibers

Let $e$ be a nilpotent element in a classical Lie algebra, let $\operatorname{Fix}$ denote the fixed-point variety associated with the Springer fiber, and let $S_e$ be the specified set of centrally extended orbits. Forgetting the central extensions gives a set $R_e$ of ordinary $A_e$-orbits. Write $\operatorname{Con}(\operatorname{Fix})$ for the set of connected components of $\operatorname{Fix}$, viewed as an $A_e$-set. Fixed-component completeness conjecture. The set $R_e$ gives the complete list, up to isomorphism, of $A_e$-orbits that appear in $\operatorname{Con}(\operatorname{Fix})$. The source notes that all $A_e$-orbits in $R_e$ already appear, so the conjecture is the assertion that there are no additional component-orbit types; the supplied passage gives no resolution.