Lusztig–Vogan natural bijection conjecture

Let $\mathbb X_+$ be the set of dominant weights. Let $\mathcal N_\mathbb C$ be the nilpotent cone of the complex reductive group $G_\mathbb C$, and let $\Xi_{G_\mathbb C}$ be the set of pairs consisting of a nilpotent $G_\mathbb C$-orbit and an isomorphism class of an indecomposable tilting vector bundle on that orbit.

Lusztig–Vogan natural bijection conjecture. There is a natural bijection

$$\mathbb X_+\overset{\sim}{\longleftrightarrow}\Xi_{G_\mathbb C}.$$

The statement was independently predicted by Lusztig and Vogan around 1990. The supplied text does not state whether it has been resolved.