The group-algebra and representation-count polynomiality conjecture for 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 a partition $\lambda\in\Lambda$, let $M_\lambda$ be the corresponding finite $\mathfrak{o}$-module and $G_\lambda=\operatorname{Aut}_{\mathfrak{o}}(M_\lambda)$, and let $f_m=f_m^{\lambda,\mathfrak{o}}$ denote the number of irreducible representations of $G_\lambda$ of dimension $m$. Group-algebra and representation-count conjecture. The isomorphism type of the group algebra $\mathbb{C}G_\lambda$ depends only on $\lambda$ and $q$. Moreover, $f_m=f_m^\lambda$ belongs to $\mathbb{Q}[q]$. The strong version is proved in the source for all $\lambda\in\Lambda_2$, while the general assertion remains open.