Properly discontinuous group actions on affine homogeneous spaces

Let $G$ be a real algebraic group, $H \leq G$ an algebraic subgroup containing a maximal reductive subgroup of $G$, and $\Gamma $ a subgroup of $G$ acting on $G/H$ by left translations. We conjecture that $\Gamma $ is virtually solvable provided its action on $G/H$ is properly discontinuous and $\Gamma \backslash G/H$ is compact, and we confirm this conjecture when $G$ does not contain simple algebraic subgroups of rank ${\geq }\,2$. If the action of $\Gamma $ on $G/H$ (which is isomorphic to an affine linear space $\mathbb A^n$) is linear, our conjecture coincides with the Auslander conjecture. We prove the Auslander conjecture for $n\leq 5$.