Zimmer's finite-extension conjecture in the critical dimension

Let $G$ be a higher-rank semisimple Lie group, let $\Gamma$ be a higher-rank lattice in $G$, and let $r(G)$ be the root-theoretic dimension defined in the source. Let $M$ be a manifold with

$$\dim(M)=r(G).$$

Given a sufficiently smooth action $\alpha$ of $\Gamma$ on $M$, Zimmer's critical-dimension conjecture. Either there exists an $\alpha$-invariant Borel probability measure on $M$, or there is a maximal parabolic subgroup $Q\subset G$ such that $M$ is diffeomorphic to a finite cover of $Q\backslash G$; moreover, $\alpha$ is smoothly conjugate to a lift of the standard right action of $\Gamma$ on $Q\backslash G$. The preceding theorem proves a measurable finite-extension alternative in the critical dimension, while this conjecture asks for the stronger topological and smooth classification. The supplied source gives no resolution.