Generic stability conjecture for definable groups over valuation rings

Let ${\mathcal O}_\mathbb{K}$ be the valuation ring of a field with residue map $\operatorname{res}$, and let $G\subseteq {\mathcal O}_\mathbb{K}^n$ be a $\emptyset$-definable group. Write $G^{00}$ for the smallest type-definable subgroup of bounded index, and let $S_G(\mathbb{K})$ denote the space of types on $G$ over $\mathbb{K}$. For a type $p\in S_G(\mathbb{K})$, let $\operatorname{res}(p)$ be its image under the residue map. Generic stability conjecture. (i) $G$ is generically stable. (ii) $G^{00}$ is a finite-index subgroup of $G$. (iii) If $p\in S_G(\mathbb{K})$ is generically stable, then $p$ is dominated by $\operatorname{res}(p)$. This conjecture extends the preceding results for linear algebraic groups over ${\mathcal O}_\mathbb{K}$, such as $\mathrm{SL}(n,{\mathcal O}_\mathbb{K})$ and $\mathrm{SO}(n,{\mathcal O}_\mathbb{K})$. It predicts generic stability, finite index of the connected component, and residue domination for arbitrary $\emptyset$-definable groups of this form; the supplied text does not state a resolution.