Openness conjecture for proper and cocompact actions

Let $G/H$ be a homogeneous space of reductive type, and let $\Gamma$ be a discrete subgroup of $G$ acting properly discontinuously and cocompactly on $G/H$. Write $\operatorname{Hom}(\Gamma,G)$ for the space of representations of $\Gamma$ in $G$. Openness conjecture. There is a neighborhood $\mathcal{U}\subset\operatorname{Hom}(\Gamma,G)$ of the natural inclusion such that every $\rho\in\mathcal{U}$ is discrete and faithful, and the action of $\Gamma$ on $G/H$ via $\rho$ is properly discontinuous and cocompact. Stability under small deformations would give openness of the space of proper and cocompact actions, a phenomenon known in important special cases but posed here in this generality.