The torus analogue of Mazur's conjecture

Let $S$ be an algebraic anisotropic torus defined over ${\mathbb Q}$, let $F$ be a finitely generated subgroup of $S({\mathbb Q})$, and let $\Gamma$ be any arithmetic subgroup of $S({\mathbb Q})$. Write $\Gamma_T=\Gamma\cap T({\mathbb Q})$ for a subtorus $T$ of $S$ defined over ${\mathbb Q}$.

The torus analogue of Mazur's conjecture. The connected component of the identity of the closure of the image of $F$ in $\Gamma\backslash S({\mathbb R})$ equals the connected component of the identity of $\Gamma_T\backslash T({\mathbb R})$ for some such $T$. Equivalently, the closure of the image of $F$ is dense in the identity component of $\Gamma\backslash S({\mathbb R})$ if and only if every subgroup $G$ of $S({\mathbb Q})$ surjecting onto the image of $F$ in $\Gamma\backslash S({\mathbb R})$ is Zariski dense in $S$.

This is the paper's central conjecture about closures of finitely generated rational subgroups in compact quotients of anisotropic tori. The source proves it assuming Schanuel's conjecture, while noting that an unconditional proof may follow from more elementary geometry-of-numbers methods.