Kazhdan–Lusztig singular-support conjecture for Schubert varieties

Let $X$ be the flag manifold for $SL(n)$, let $X_w$ be the Schubert variety corresponding to $w$ in the Weyl group $W$, and let $\sideset{^{\pi}}{_{X_w}}{\Bbb C}$ denote the associated perverse sheaf. Its singular support is a closed Lagrangian subvariety of $T^*X$.

Kazhdan–Lusztig singular-support conjecture. The singular support of $\sideset{^{\pi}}{_{X_w}}{\Bbb C}$ is irreducible.

This conjecture is presented as following from, and being equivalent to, the preceding problem for type $A$. The supplied text does not give a resolution.