Finite-cover descent conjecture for similarity of matrices over discrete valuation rings

Let $R$ be a discrete valuation ring (DVR) with fraction field $K$, and let $A$ and $B$ be $n\times n$ matrices over $R$. Suppose that $L/K$ is a finite extension and $S$ is the integral closure of $R$ in $L$. Finite-cover descent conjecture. If $A$ and $B$ are similar over $S$, then $A$ and $B$ are similar over $R$. The conjecture asks whether similarity over a finite integral extension of a discrete valuation ring descends to the original ring. The analogous assertion for similarity over the fraction field is false in general, while the conjecture is verified in the paper for $2\times2$ matrices and for matrices with separable characteristic polynomials.