Similarity classes over finite quotients and module isomorphism classes

Let $R$ be a discrete valuation ring with residue field $\mathbf F_q$, let $P$ be its maximal ideal, and let $M_m(R/P^n)$ denote the ring of $m\times m$ matrices over $R/P^n$. An $m$-dimensional $\mathbf F_q[x_1,\dotsc,x_n]$-module is a module whose underlying $\mathbf F_q$-vector space has dimension $m$. Similarity–module-counting conjecture. The number of similarity classes in $M_m(R/P^n)$ is equal to the number of isomorphism classes of $m$-dimensional $\mathbf F_q[x_1,\dotsc,x_n]$-modules for all positive integers $m$ and $n$. The equality is known in several low-dimensional cases, including $m\leq 3$, but the general statement is left as a conjecture and concerns a connection between matrix similarity over finite local rings and representations of polynomial algebras.