McNamara's center conjecture for cyclotomic Hecke algebras

Let $K$) be a field and let $1\neq \xi\in K^\times$. Write

$$\Lambda=\Lambda_{\kappa_1}+\cdots+\Lambda_{\kappa_r},\qquad \kappa_i\in\mathbb{Z}\quad(1\leq i\leq r),$$

and let $\mathscr{H}_{n,K}^\Lambda$ be the corresponding cyclotomic Hecke algebra, with Jucys–Murphy elements $\mathcal{L}_1,\ldots,\mathcal{L}_n$.

McNamara's center conjecture. The center $Z(\mathscr{H}_{n,K}^\Lambda)$ is precisely the set of symmetric polynomials in $\mathcal{L}_1,\ldots,\mathcal{L}_n$.

The conjecture identifies the center with the symmetric-polynomial subalgebra generated by the commuting Jucys–Murphy elements. It was proved in several special cases, but the source states that it remains open in general.