Kobayashi's standard quotient conjecture for reductive homogeneous spaces

Let $G/H$ be a reductive homogeneous space. A compact quotient is a quotient $\Gamma\backslash G/H$ by a discrete subgroup $\Gamma$ acting properly discontinuously and cocompactly; a standard compact quotient is one arising from a reductive subgroup acting transitively on $G/H$ with compact stabilizer.

Kobayashi's conjecture. The reductive homogeneous space $G/H$ admits compact quotients if and only if it admits standard ones.

This conjecture asks whether every compact quotient of a reductive homogeneous space is standard up to deformation or other constructions. It remains open.