Kobayashi's rank conjecture on Clifford–Klein forms

A Clifford–Klein form of a homogeneous space $G/H$ is a quotient space $\Gamma \backslash G/H$, where $\Gamma$ is a discrete subgroup of $G$ acting properly and freely on $G/H$. The homogeneous space $G/H$ is of reductive type if $G$ is a linear reductive Lie group with Cartan involution $\theta$ and $H$ is a closed subgroup of $G$ with finitely many connected components such that $\theta(H)=H$. Let $K=G^\theta$ and $K_H=H^\theta$ be the corresponding maximal compact subgroups, and let $\operatorname{rank}$ denote complex rank. Kobayashi's rank conjecture. A homogeneous space $G/H$ of reductive type does not admit a compact Clifford–Klein form if

$$\operatorname{rank} G-\operatorname{rank} K<\operatorname{rank} H-\operatorname{rank} K_H.$$

The conjecture concerns a nonexistence obstruction for compact Clifford–Klein forms, which are manifolds locally modelled on $G/H$. The paper states that the conjecture is solved affirmatively using a cohomological obstruction and the Sullivan model for a reductive pair.