McGovern's character formula conjecture for nilpotent orbit coordinate rings

Let $G$ be a complex reductive group, let $W_G$ be its Weyl group, and let $\mathcal{O}$ be a nilpotent orbit. Write $R(\mathcal{O})$ for the $G$-module of regular functions on $\mathcal{O}$, and let $Ind_T^G(\lambda)$ denote the induced representation associated with a character $\lambda$.

McGovern's conjecture. For each nilpotent orbit $\mathcal{O}$, there are a fixed character $\mu$, rational numbers $c_w$, and a subset $W_{\mathcal{O}}$ of $W_G$ such that

$$R(\mathcal{O}) \cong \sum_{w \in W_{\mathcal{O}}} c_w Ind_T^G(\mu-w\cdot\mu).$$

The conjecture gives a uniform Weyl-group expression for the $G$-structure of regular functions on nilpotent orbits. The paper proves it for the orbits $(2^{2p}1^{2q})$ in $Sp(2n,\mathbb{C})$ and for their simply connected covers; its status beyond these cases is not specified here.