Modified Onshuus–Pillay conjecture with a small-model-definable subgroup

Let $M$ be a highly saturated $P$-minimal structure over a $p$-adic field $K$, let $G$ be a definable group, and set $n=\dim(G)$. Let $M_0\prec M$ be a small model defining $G$. Recall that $G^{00}$ is the smallest type-definable subgroup of $G$ of small index, and that $H$ is compactly dominated by $H/H^{00}$ when the quotient map satisfies the compact-domination condition with respect to normalized Haar measure. Modified Onshuus–Pillay conjecture. There is an $n$-dimensional $M_0$-definable open subgroup $H\subseteq G$ such that

$$H/H^{00}$$

is isomorphic to an $n$-dimensional Lie group over $K$, and $H$ is compactly dominated by $H/H^{00}$. This strengthens the original formulation by requiring the subgroup to be definable over the fixed small model $M_0$. The source says that the original Onshuus–Pillay conjecture is technically resolved, but that the author does not know how to prove this modified variant; it is therefore open.