Achar–Sommers' conjecture on regular functions of nilpotent orbit covers

Let $\mathfrak{g}$ be a classical Lie algebra, let $\mathcal{N}_o$ be its nilpotent orbits, and let ${}^L\mathcal{N}_o$ be the nilpotent orbits in the Langlands dual Lie algebra. Let $\mathcal{O}^{\vee}$ be a classical nilpotent orbit in ${}^L\mathcal{N}_o$ with canonical preimage $(\mathcal{O},C)$ under Sommers' map. Let $G_C$ be the subgroup corresponding to $C$, and write

$$R(\widetilde{\mathcal{O}}^C)\cong \operatorname{Ind}_{G_C}^{G}(\mathrm{triv})=\sum_{\lambda\in\Lambda^+}m_\lambda\operatorname{Ind}_{T}^{G}(\lambda).$$

Achar–Sommers' conjecture. The maximal element in this expression is $h^{\vee}$, the semisimple element of a Jacobson–Morozov triple of $\mathcal{O}^{\vee}$. The paper states that its calculations prove this conjecture for the relevant classical special cases; the parser supplies no formal resolution status for the conjecture itself.