The co-amenable subgroup conjecture for higher-rank semisimple groups

Let $G$ be a semisimple Lie group without compact factors and of rank at least two. A discrete subgroup $\Lambda\le G$ is co-amenable when the Hilbert space $L^2(G/\Lambda)$ admits $G$-asymptotically invariant vectors. The co-amenable subgroup conjecture. If $\Lambda\le G$ is discrete, co-amenable, and projects densely to every proper factor of $G$, then $\Lambda$ is a lattice in $G$. This conjecture is stated as a general conjecture implying the Stuck--Zimmer conjecture. The source gives the higher-rank co-amenability result for irreducible invariant random subgroups, while the asserted conclusion for all such discrete subgroups remains open there.