Chavli–Pfeiffer's integral basis conjecture for Hecke algebra centers

Let $W$ be a finite group with Hecke algebra $\mathscr{H}(W)$ over a ring $R$, and let $\{\mathbf{b}_w\mid w\in W\}$ be an $R$-basis. After extending scalars to the field $F$, let $\{\mathbf{b}_w^\vee\mid w\in W\}$ be the dual basis with respect to the symmetrizing form, and define coefficients $g_{w,C}$ by

$$\mathbf{b}_w^\vee\equiv\sum_{C\in\operatorname{Cl}(W)}g_{w,C}\mathbf{b}_{w_C}^\vee\pmod{[F\otimes_R\mathscr{H}(W),F\otimes_R\mathscr{H}(W)]}.$$

Set

$$z_C=\sum_{w\in W}g_{w,C}\mathbf{b}_w.$$

Here $w_C$ is a representative of the conjugacy class $C$.

Chavli–Pfeiffer's integral basis conjecture. There exist an $R$-basis $\{\mathbf{b}_w\mid w\in W\}$ of $\mathscr{H}(W)$ and conjugacy-class representatives $\{w_C\mid C\in\operatorname{Cl}(W)\}$ such that $g_{w,C}\in R$ for every pair $(w,C)$, and consequently $\{z_C\mid C\in\operatorname{Cl}(W)\}$ is an $R$-basis of $Z(\mathscr{H}(W))$.

Over the fraction field, the analogous elements form a basis of the center by the cited Chavli–Pfeiffer result. The conjecture asks for choices making this basis integral over $R$; the source does not state a general resolution.