Kobayashi's reductive constructor conjecture for compact Clifford–Klein forms

Let $G/H$ be a homogeneous space of reductive type. It admits a reductive constructor when it satisfies the constructor condition of the paper, namely that $G/H$ admits a reductive constructor.

Reductive constructor conjecture. If the homogeneous space $G/H$ of reductive type admits compact Clifford–Klein forms, that is, $G/H$ admits a reductive constructor.

This conjecture concerns the existence problem for compact Clifford–Klein forms of pseudo-Riemannian homogeneous spaces. The constructor idea for the reductive case was introduced by T. Kobayashi, but the conjecture remains open.