Polynomial dimension-and-frequency conjecture for representations of general linear groups over local rings

Let $\mathfrak{O}$ be the set of rings which are the valuation ring of a non-Archimedean local field, up to isomorphism. Fix integers $n,r\geq 1$, and let $\operatorname{Irr}(G)$ denote the irreducible complex representations of a group $G$ up to isomorphism. For $\mathfrak{o}\in\mathfrak{O}$, write $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ and $\mathfrak{o}_1=\mathfrak{o}/\mathfrak{p}$.

Polynomial dimension-and-frequency conjecture. There exist $k\geq 1$, polynomials

$$d_1(x),\dots,d_k(x)\in\mathbb{Z}[x]\setminus\{0\},\qquad m_1(x),\dots,m_k(x)\in\mathbb{Q}[x]\setminus\{0\},$$

and, for every $\mathfrak{o}\in\mathfrak{O}$ and $i\in\{1,\dots,k\}$, subsets $\mathcal{R}_{i,\mathfrak{o}}\subseteq\operatorname{Irr}(\mathrm{GL}_n(\mathfrak{o}_r))$ such that

$$\bigcup_{i=1}^k\mathcal{R}_{i,\mathfrak{o}}=\operatorname{Irr}(\mathrm{GL}_n(\mathfrak{o}_r))$$

is a disjoint union, with

1. for all $\rho\in\mathcal{R}_{i,\mathfrak{o}}$, $\dim\rho=d_i(|\mathfrak{o}_1|)$; and
2. $|\mathcal{R}_{i,\mathfrak{o}}|=m_i(|\mathfrak{o}_1|)$.

This is the more precise restatement proposed by the authors of Onn's conjecture that representation dimensions and their frequencies are polynomial in the residue-field cardinality. The source gives no evidence resolving it.