Vogan's representation conjecture for real reductive groups

Let $K$ be a symmetric subgroup of $G$, let $\OO$ be a nilpotent orbit satisfying

$$\operatorname{codim}(\partial \OO,\overline{\OO}) \geq 4,$$

and let $\cB$ be an irreducible Harish-Chandra $(\cA_0^{\widetilde{X}},K)$-module. Let $\OO_{\cB}$ be the associated $K$-orbit and let $\mathcal{L}_{\cB}$ be Vogan's twisted local system on it.

Vogan's conjecture. There is an isomorphism of $K$-representations

$$\cB \simeq_K \Gamma(\OO_{\cB},\mathcal{L}_{\cB}).$$

The conjecture is equivalent to the isomorphism of a natural embedding whose cokernel is supported on the boundary. The source reports that the cokernel has support of codimension at least $3$ in forthcoming work, but does not establish the conjecture itself.