Reductive projection conjecture for compact Clifford–Klein forms

Let $G$ be a connected real linear reductive Lie group, let $H$ be a reductive subgroup of $G$, and let $\Gamma$ be a discrete subgroup of $G$. Let the Zariski closure of $\Gamma$ have Levi decomposition $L\ltimes U$, and let $\pi_{\operatorname{red}}:L\ltimes U\to L$ be the projection to the Levi factor. Reductive projection conjecture. If $\Gamma$ acts properly discontinuously and cocompactly on $G/H$, then the restriction $\pi_{\operatorname{red}}|_\Gamma$ is discrete and faithful. This would reduce compact quotient questions to reductive Zariski closures; the paper establishes the consequence under the stated discreteness and faithfulness hypothesis, while the conjecture asserts that this hypothesis is automatic.