Tomanov's crystallographic actions conjecture for real algebraic homogeneous spaces

Let $G$ be a real algebraic group, let $H\subgroup G$ contain a maximal reductive subgroup of $G$, and let $\Gamma\subgroup G$. Say that $\Gamma$ acts crystallographically on $G/H$ when it is discrete, acts properly on $G/H$, and the quotient $\Gamma\backslash G/H$ is compact. Tomanov's conjecture. If $\Gamma$ acts crystallographically on $G/H$, then $\Gamma$ is virtually polycyclic.

This generalizes Auslander's conjecture from affine space to homogeneous spaces of real algebraic groups. The paper states that Tomanov proved this stronger statement for dimensions at most five, while the general case remains open.