Kobayashi's standardness conjecture for compact Clifford–Klein forms

Let $G$ be a connected semisimple non-compact Lie group and let $H$ be a closed subgroup. A Clifford–Klein form of $G/H$ is a quotient $\Gamma\backslash G/H$ by a discrete subgroup $\Gamma\subset G$ acting properly and cocompactly. It is standard if it arises from a standard triple $(G,H,L)$, where $G=HL$ for reductive subgroups $H,L\subset G$ and $H\cap L$ is compact.

Kobayashi's conjecture. If $G/H$ has a compact Clifford–Klein form, then there is a structure of a Clifford–Klein form on $G/H$ which is standard.

The conjecture does not assert that every compact Clifford–Klein form is standard, only that a standard structure should exist for the underlying homogeneous space. Many partial results support it, and all known examples are standard or arise by deforming the discrete subgroup in a standard construction, but the general question remains open.