Cohomological realization of the Lusztig–Vogan bijection

Suppose we are in the reductive case. Let $\boldsymbol{ u}\in-\boldsymbol{X}^+$, and let $({\mathscr{O}}_{\boldsymbol{ u}},\boldsymbol{\rho}_{\boldsymbol{ u}})\in\Xi$ be the pair corresponding to $\boldsymbol{ u}$ under the Lusztig–Vogan bijection. Let $\mathsf{T}(w_{\boldsymbol{ u}}\cdot_\ell 0)$ be the corresponding tilting module and let $\mathsf{H}_{\mathsf{g}}^\bullet$ denote its coherent cohomology sheaf. Cohomological realization conjecture. The coherent sheaf $\mathsf{H}_{\mathsf{g}}^\bullet(\mathsf{T}(w_{\boldsymbol{ u}}\cdot_\ell 0))$ is scheme-theoretically supported on $\overline{{\mathscr{O}}_{\boldsymbol{ u}}}$, and

$$\mathsf{H}_{\mathsf{g}}^\bullet(\mathsf{T}(w_{\boldsymbol{ u}}\cdot_\ell 0))|_{{\mathscr{O}}_{\boldsymbol{ u}}}\cong \mathscr{T}_{{\mathscr{O}}_{\boldsymbol{ u}}}(\boldsymbol{\rho}_{\boldsymbol{ u}}).$$

This would provide a faithful reductive analogue of the corresponding cohomological theorem and would solve the cohomology problem mentioned by the authors. The supplied source gives no evidence of resolution.