Center conjecture for cyclotomic KLR algebra weight spaces

Let $\Lambda=\sum_{i\in I}\lambda_i\Lambda_i\in X^+$ and let $R^\Lambda(\alpha)$ be the summand of the cyclotomic KLR algebra corresponding to $\alpha\in I^m$. Assume that $\alpha$ corresponds to a nonzero weight space of the irreducible representation $V(\Lambda)$. The ring $R^\Lambda(\alpha)$ is graded, and its center is therefore graded. Center conjecture. The center of $R^\Lambda(\alpha)$ is zero dimensional in negative degrees and one dimensional in degree zero. The source gives no evidence of a resolution for this assertion; it appears after the cyclotomic quotient conjecture and concerns the graded centers of its weight-space summands.