Double-coset characterization conjecture for simple-group actions

Consider the double-coset $G$-spaces of the form $K\backslash L/\Gamma$, where $L$ is a semisimple Lie group with an irreducible lattice $\Gamma$ and a compact subgroup $K$, and the $G$-action is induced by a nontrivial homomorphism $G\rightarrow L$ whose image centralizes $K$. A $G$-action on a manifold $M$ is equivalent to a double-coset action conjecture of this form for some $L$, $\Gamma$, and $K$ if and only if it has a dense orbit, preserves a pseudo-Riemannian metric, and preserves a transverse geometric structure to the orbits suitably related to the geometry of $GK\backslash L$. The conjecture is intended to provide a geometric characterization of the double-coset examples of $G$-actions; the precise meaning of “suitably related” and the general validity of the equivalence remain open.