Finite generation conjecture for intermediate groups of $SL_2$ over higher-dimensional domains

Let $R$ be a finitely generated integral domain of Krull dimension at least $3$, let $F$ be its field of fractions, and let $\Gamma$ be a group satisfying

$$SL_2(R)\leq \Gamma\leq SL_2(F).$$

Finite generation conjecture. The group $\Gamma$ is never finitely generated.

This would extend the preceding non-finite-generation results from polynomial rings and related rings to all finitely generated integral domains of Krull dimension at least $3$. The source presents it as a conjectural strengthening; no resolution is given there.