Zimmer's projective-action classification conjecture

For $n\geq 3$, let $\Gamma\subset\mathrm{SL}(n,\mathbb{R})$ be a lattice, let $M$ be a closed $(n-1)$-dimensional manifold, and let $\alpha:\Gamma\to\mathrm{Diff}^{\infty}(M)$ be an action with infinite image.

Zimmer's conjecture. Either $M=S^{n-1}$, or $M=\mathbb{R}P^{n-1}$ and $\alpha$ is $C^{\infty}$-conjugate to the projective action on $S^{n-1}$ or $\mathbb{R}P^{n-1}$.

This conjecture is motivated by measurable classification results showing that non-measure-preserving critical-dimensional actions have the projective action as a measurable factor. It remains open as a smooth classification statement.