Polynomiality conjecture for almost-character scalar products

Let $\mathbb{G}=(W,\Lambda)$ be a $\mathbb{Z}_\ell$-reflection group with $(|W|,\ell)=1$, and let $T_a=\Lambda/\ell^a\Lambda$. For $b\ge a$, regard each $\psi\in\operatorname{Irr}(T_a)$ as a character of $T_b$ by inflation. Let $f_\varphi^{C_{\mathbb{G}}(t)}$ denote the relevant fake-degree polynomial and $R_\varphi$ the almost character indexed by $\varphi\in\operatorname{Irr}(W)$. Polynomiality conjecture. For all $a\ge1$, $\varphi\in\operatorname{Irr}(W)$, and $\psi\in\operatorname{Irr}(T_a)$, there is a polynomial $F_{a,\varphi,\psi}\in\mathbb{Z}[y]$ with non-negative coefficients such that, for every $b\ge a$,

$$\langle R_\varphi,\psi\rangle_{T_b}:=|T_b|^{-1}\sum_{t\in T_b}\psi(t)\left(f_\varphi^{C_{\mathbb{G}}(t)}\right)_{x=\ell^b+1}=F_{a,\varphi,\psi}(\ell^b).$$

This extends the almost-character integrality conjecture in the coprime reflection-group setting and is presented as an expected strengthening; no general proof is given.