A conjugacy theorem for subgroups of $\mathrm{SL}_n$ containing the group of diagonal matrices

Let $R$ be a commutative local ring. It is proved that if $n\ge3$ and the residue field of $R$ contains at least $3n+2$ elements, then the subgroup of diagonal matrices in the special linear group of degree $n$ over $R$ is pronormal. For semilocal rings with the same restrictions on residue fields this subgroup is paranormal.