The center conjecture for the parabolic category $\mathcal{O}^{n,n}$

Let $\mathcal{O}^{n,n}$ be the parabolic highest weight category for $\mathfrak{sl}_{2n}$, and let $A_{n,n}$ be its finite-dimensional algebra model. Let $e\in A_{n,n}$ be the idempotent for which $h:H^n\otimes_{\mathbb Z}\mathbb C\cong eA_{n,n}e$. The center conjecture.

$$Z(A_{n,n})\cong Z(eA_{n,n}e),$$

and $h$ induces an isomorphism between the centers of $H^n\otimes_{\mathbb Z}\mathbb C$ and $\mathcal{O}^{n,n}$. The claim predicts that the center is unchanged on passing to the idempotent corner; the source gives no resolution.