Conjecture on syndetic conjugates of a Cartan subgroup

Let $G$ be a definably simple group definable in an o-minimal structure $\mathcal{M}$. A subset $X\subseteq G$ is syndetic if there is a finite set $E\subseteq G$ such that $XE=G$. For a subgroup $H\leq G$, write $H^G$ for the union of its conjugates in $G$. Cartan subgroup conjugacy conjecture. There is a Cartan subgroup $H$ of $G$ such that $H^G$ is syndetic in $G$. This extends the known example of $\operatorname{SL}(2,\mathbb{R})$, where the conjugates of the diagonal Cartan subgroup form a syndetic subset; whether the assertion holds for every definably simple group definable in an o-minimal structure remains open.