Property $R_\infty$ for finitely generated special linear groups over integral domains

Let $n\geq 4$ and let $I\subseteq\{1,\dots,n\}$ satisfy the (NG)-condition. For an infinite integral domain $R$, write $\mathbf{S}^I_n(R)$ for the corresponding group, and assume that it is finitely generated. Property $R_\infty$ conjecture. If $R$ has arbitrary characteristic, then $\mathbf{S}^I_n(R)$ has property $R_\infty$. The result is motivated by the known positive-characteristic cases and by the search for infinitely many fixed points in $R$ for every ring automorphism; establishing the claim for arbitrary infinite integral domains remains open.