Finite-polynomial parametrisation conjecture for modular character degrees

Let $G$ be a finite group of Lie type with Weyl group ${\mathbf{W}}$ and Frobenius-induced automorphism $\gamma$, defined over a field with parameter $q$, and let $k$ be a field of characteristic $\ell>0$. Write $\operatorname{Irr}_k(G)$ for the irreducible $kG$-modules and $\mathbb K[t]$ for the polynomial ring in $t$. Finite-polynomial parametrisation conjecture. There exists a finite set of polynomials $\bar{\mathcal D}({\mathbf{W}},\gamma)\subseteq\mathbb K[t]$, depending only on ${\mathbf{W}}$ and $\gamma$ and not on $q$ or $\ell$, such that

$$\{\dim Y\mid Y\in\operatorname{Irr}_k(G)\}\subseteq\{f(q)\mid f\in\bar{\mathcal D}({\mathbf{W}},\gamma)\}.$$

This would give a uniform finite list of polynomial expressions for all irreducible modular character degrees across the relevant groups and characteristics; the supplied text gives no resolution status.