Onn's weak conjecture on representations of general linear groups over local rings

Let $F$ and $F'$ be non-Archimedean local fields with rings of integers $\mathfrak{o}$ and $\mathfrak{o}'$ having the same finite residue cardinality $q$. For fixed integers $n,r\geq 1$, write $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ and $\mathfrak{o}'_r=\mathfrak{o}'/(\mathfrak{p}')^r$, and let $\operatorname{Irr}(G)$ denote the irreducible complex representations of a group $G$ up to isomorphism.

Onn's weak conjecture. There is an isomorphism of group algebras

$$\mathbb{C}[\mathrm{GL}_n(\mathfrak{o}_r)]\cong\mathbb{C}[\mathrm{GL}_n(\mathfrak{o}'_r)],$$

or equivalently, a dimension-preserving bijection

$$\operatorname{Irr}(\mathrm{GL}_n(\mathfrak{o}_r))\longleftrightarrow\operatorname{Irr}(\mathrm{GL}_n(\mathfrak{o}'_r)).$$

This conjecture asserts that the complex representation dimensions of these general linear groups depend only on the residue cardinality, not on the local field. It is presented as a conjecture of Onn; the source does not state a resolution.