The orbit-method quantization conjecture for nilpotent orbit covers

Let $G$ be a complex simple Lie group with Lie algebra $\mathfrak{g}$, and let $K$ be a maximal compact subgroup of $G$ with complexification $K_{\mathbb{C}}$. For a nilpotent orbit $\mathcal{O}$, let $R(\mathcal{O})$ be its ring of regular functions. For $e ein\mathcal{O}$, let $G_e$ be its isotropy group, let $(G_e)^0$ be the identity component, and define $A(\mathcal{O})=G_e/(G_e)^0$. For an irreducible representation $\rho$ of $A(\mathcal{O})$, let $R(\mathcal{O},\rho)$ denote the global sections of the $G$-equivariant bundle $G\times_{G_e}V_{\rho}\to G/G_e\cong\mathcal{O}$. The orbit-method quantization conjecture. There \exists a (not necessarily unique) $(\mathfrak{g}_{\mathbb{C}},K_{\mathbb{C}})$-module $Q$ such that

$$Q|_{K_{\mathbb{C}}}\cong R(\mathcal{O}),$$

and, more generally, for every irreducible representation $\rho$ of $A(\mathcal{O})$, there exists a $(\mathfrak{g}_{\mathbb{C}},K_{\mathbb{C}})$-module $Q_{\rho}$ such that

$$Q_{\rho}|_{K_{\mathbb{C}}}\cong R(\mathcal{O},\rho)=\operatorname{Ind}_{G^e}^{G}(\rho).$$

When $\rho$ is trivial, $R(\mathcal{O},\rho)=R(\mathcal{O})$. This is the orbit-method prediction for quantizing nilpotent orbits and their equivariant local systems; the source gives no resolution status.