Vogan's conjecture for complex nilpotent coadjoint orbits

Let $G$ be a complex reductive Lie group, and let $X=G\cdot\lambda=G/G(\lambda)$ be a nilpotent coadjoint orbit. Suppose an irreducible $G$-equivariant local system on $X$ is given; equivalently, suppose there is an irreducible representation $\chi$ on $V_{\chi}$ of the finite group $G(\lambda)/G(\lambda_0)$, or an indecomposable $G$-equivariant holomorphic vector bundle $\mathcal{V}_{\chi}$ on $X$ with flat connection. Vogan's conjecture. There is attached to $\chi$ an irreducible unitary representation $\pi(\lambda,\chi)$ of $G$ whose space of $K$-finite vectors is isomorphic to the space of algebraic sections of the bundle $\mathcal{V}_{\chi}$. This is the complex-group reduction of the preceding conjecture and is verified in the paper for the model orbit in type $G_2$ of complex dimension $8$.