The center isomorphism conjecture for parabolic algebras

Let $A_{\kappa}$ be a finite-dimensional algebra with $A_{\kappa}\text{-mod}\cong\mathcal{O}^{\kappa}$, and let $e\in A_{\kappa}$ be the maximal idempotent such that the left $A_{\kappa}$-module $A_{\kappa}e$ is injective. The center isomorphism conjecture. Inclusion $eA_{\kappa}e\subset A_{\kappa}$ induces an isomorphism

$$Z(A_{\kappa})\cong Z(eA_{\kappa}e).$$

This generalizes the preceding center claim from $A_{n,n}$ to arbitrary parabolic algebras; the source gives no resolution.