The center conjecture for cyclotomic KLR algebras

Let $\beta\in Q_n^+$ and let $\Lambda\in P^+$. Let $p_\beta^\Lambda:\mathscr{R}_\beta\twoheadrightarrow\mathscr{R}_\beta^\Lambda$ be the canonical surjection. The center $Z(\mathscr{R}_\beta)$ consists of the elements fixed by the natural symmetric-group action on the KLR generators $x_1,\ldots,x_n$ and $e(\nu)$, for $\nu\in I^\beta$. The center conjecture. The induced map

$$p_\beta^\Lambda\big|_{Z(\mathscr{R}_\beta)}:Z(\mathscr{R}_\beta)\longrightarrow Z(\mathscr{R}_\beta^\Lambda)$$

is always surjective. In particular, $Z(\mathscr{R}_\beta^\Lambda)$ consists of symmetric elements in its KLR $x$ and $e(\nu)$ generators. The conjecture describes the relation between the centers of KLR algebras and their cyclotomic quotients; the source identifies it as an unsolved open problem.