The center conjecture for cyclotomic KLR algebras of type A

Let $\alpha\in Q_n^+$ and let $\Lambda=\Lambda_{\kappa_1}+\cdots+\Lambda_{\kappa_r}\in P^+$, where $\kappa_1,\ldots,\kappa_r\in\mathbb{Z}$. Let $\mathscr{R}_{\alpha,K}^{\Lambda}$ be the corresponding cyclotomic KLR algebra, and call an element symmetric when it is the image of an element fixed by the natural place-permutation action of $\mathfrak{S}_n$ on the polynomial-idempotent subalgebra.

Center conjecture. The center of $\mathscr{R}_{\alpha,K}^{\Lambda}$ consists precisely of the symmetric elements in $\mathscr{R}_{\alpha,K}^{\Lambda}$.

This is the KLR-algebra formulation of the center conjecture for type $A$. The source presents it as a conjecture without giving a general resolution.