Transitivity conjecture for centralizers on reduction-type fibers

Let $\underline{\mathcal N}$ be the set of nilpotent orbits and let $\operatorname{KL}:\underline{\mathcal N}\to\underline W$ be the Kazhdan–Lusztig map. For $\mathcal O\in\underline{\mathcal N}$, set $[w]=\operatorname{KL}(\mathcal O)$, and let $\mathfrak g\in(L^{\heartsuit}\mathfrak g)^{\mathrm{sh}}_{[w]}$. Let $\operatorname{Gr}_{\mathfrak g,\mathcal O}$ be the reduction-type fiber and $(LG)_{\mathfrak g}$ the centralizer of $\mathfrak g$ in $LG$.

Centralizer transitivity conjecture. The group $(LG)_{\mathfrak g}$ acts transitively on $\operatorname{Gr}_{\mathfrak g,\mathcal O}$. The paper notes that this fiber is nonempty and discrete under the stated hypotheses; the supplied excerpt does not establish whether the proposed transitivity is subsequently proved.