Equivalence between cocompact discontinuous groups and compact standard quotients

Prove that for homogeneous spaces G/H of reductive type, G/H admits a cocompact properly discontinuous group if and only if G/H admits a compact standard quotient, i.e., there exists a reductive subgroup L acting properly on G/H and a torsion-free cocompact lattice \Gamma \subset L such that \Gamma\G/H is compact.

Background

The conjecture aims to reduce the global classification of compact CliffordKlein forms to checking existence of proper and cocompact actions by reductive subgroups (standard quotients), for which explicit criteria are available.

It would imply that solving the original classification problem amounts to verifying finitely many representation-theoretic and geometric conditions on candidate pairs (G,H).