The standard quotient conjecture for reductive homogeneous spaces

Let $G/H$ be a homogeneous space of reductive type. A cocompact properly discontinuous group means a properly discontinuous group whose quotient of $G/H$ is compact, and a compact standard quotient is a quotient $[?]$ arising from a cocompact torsion-free discrete subgroup of a reductive subgroup acting properly on $G/H$.

Standard quotient conjecture. The homogeneous space $G/H$ admits a cocompact properly discontinuous group if and only if $G/H$ admits a compact standard quotient.

The conjecture was proposed by the author in 2001. The supplied text gives no resolution status.