The degree bound conjecture for primitive representations of rectangular finite O-modules

Let $\mathfrak{o}$ be a discrete valuation ring with maximal ideal $\mathfrak{p}$ and residue-field cardinality $q=|\mathfrak{o}/\mathfrak{p}|$. For positive integers $\ell$ and $n$, let $G_{\ell^n}=\operatorname{Aut}_{\mathfrak{o}}(M_{\ell^n})$ be the automorphism group of the corresponding rectangular finite $\mathfrak{o}$-module, and consider its primitive irreducible representations. Degree bound conjecture. The dimensions of the primitive irreducible representations of $G_{\ell^n}$ are polynomials in $\mathbb{Z}[q]$ of degree $d$ satisfying

$$d\leq \binom{n}{2}\ell.$$

The claim concerns the polynomial dependence of primitive representation dimensions on the residue-field size and gives an explicit degree bound; no resolution is supplied in the source.