The Springer cohomology conjecture for parabolic category $\mathcal{O}^{\kappa}$

Let $\kappa=(k_1,\dots,k_m)$ be a decomposition of $n$, let $\mathcal{O}^{\kappa}$ be the corresponding parabolic highest weight category for $\mathfrak{sl}_n$, and let $\mathcal{B}_{\kappa}$ be the Springer variety of complete flags stabilized by a nilpotent operator with Jordan decomposition $(k_1,\dots,k_m)$. The Springer cohomology conjecture.

$$Z(\mathcal{O}^{\kappa})\cong \mathrm{H}^{\ast}(\mathcal{B}_{\kappa},\mathbb C).$$

This proposes a geometric description of the center of every such parabolic category; the source gives no resolution.