The conjecture on polynomial class-sum bases for centers of Hecke algebras

Let $W$ be a complex reflection group, let $H$ be its associated Hecke algebra over the parameter ring $R$, and let $\operatorname{Cl}(W)$ denote the set of conjugacy classes of $W$. For a basis $\{b_w:w\in W\}$ of $H$ and representatives $\{w_C:C\in\operatorname{Cl}(W)\}$, let $g_{w,C}$ be the coefficients produced by the construction of the preceding center theorem, and let $z_C$ be the resulting central elements.

Polynomial class-sum basis conjecture. There exists a choice of a basis $\{b_w:w\in W\}$ of $H$ and a choice of conjugacy class representatives $\{w_C:C\in\operatorname{Cl}(W)\}$ such that the construction yields polynomial coefficients $g_{w,C}\in R$, and hence a basis

$$\{z_C:C\in\operatorname{Cl}(W)\}$$

of $Z(H)$.

The claim generalizes the explicitly verified rank-two exceptional cases listed in the source and seeks an integral, parameter-polynomial basis of the center for every complex reflection group. The supplied text gives no resolution evidence, so the conjecture remains open here.