Converse to the compact Clifford–Klein form criterion

Let $G\supset H$ be a pair of reductive linear Lie groups. Suppose that a compact Clifford–Klein form $\Gamma\backslash G/H$ exists. Theorem 15 states that if a reductive subgroup $L\subset G$ satisfies

$$L\pitchfork H \quad\text{and}\quad d(L)+d(H)=d(G),$$

then a compact Clifford–Klein form of $G/H$ exists.

Converse to Theorem 15. The converse of Theorem 15 also holds.

In the converse direction, the conjecture asserts that compact Clifford–Klein forms arise from reductive subgroups satisfying the criterion in Theorem 15. The source says that this weaker converse remains unsolved.