The quaternionic non-finite-generation conjecture for anisotropic groups

Let $R$ and $F$ be as in the Krull-dimension-three non-finite-generation conjecture, so $R$ is a finitely generated integral domain of Krull dimension at least $3$ and $F$ is its field of fractions. Let $D$ be a quaternion algebra over $F$, and let $\mathcal G=SL_{1,D}$ be the algebraic group of elements of reduced norm $1$. Quaternionic non-finite-generation conjecture. There is no finitely generated group $\Gamma$ satisfying

$$\mathcal G(R) \leq \Gamma \leq \mathcal G(F).$$

This is a special anisotropic-group analogue of the preceding problem and returns to the motivation from Rapinchuk's question. The source gives no resolution, so the assertion remains open.