Dimension-equality conjecture for centers of cyclotomic Hecke algebras of type $G(r,p,n)$

Let $r,p,n$ be the parameters of the cyclotomic Hecke algebras, let $K$ be the relevant specialization field, let $\mathscr{K}$ be the generic field, and let $Z(\mathscr{H}_{r,n}(K))^{(k)}$ and $Z(\mathscr{H}_{r,n}(\mathscr{K}))^{(k)}$ denote the $k$-weight components of their centers for $k\in\mathbb{Z}/p\mathbb{Z}$. Write $Z(\mathscr{H}_{r,p,n}(K))$ and $Z(\mathscr{H}_{r,p,n}(\mathscr{K}))$ for the centers of the corresponding Hecke algebras. Dimension-equality conjecture. For any $k\in\mathbb{Z}/p\mathbb{Z}$, one has

$$\dim_K Z(\mathscr{H}_{r,n}(K))^{(k)}=\dim_{\mathscr{K}}Z(\mathscr{H}_{r,n}(\mathscr{K}))^{(k)},$$

and

$$\dim_K Z(\mathscr{H}_{r,p,n}(K))=\dim_{\mathscr{K}}Z(\mathscr{H}_{r,p,n}(\mathscr{K})).$$

These equalities would establish that specialization preserves the relevant dimensions of the centers, a key step toward constructing a seminormal basis for the cyclotomic Hecke algebra of type $G(r,p,n)$. The supplied context does not state whether the conjecture is proved or remains open.