Zimmer's volume-preserving low-dimensional action conjecture

Let $G$ be a semisimple Lie group with property $(T)$ and let $\Gamma$ be a lattice in $G$. Let $d$ be the least dimension of an infinite-image linear representation and $n$ the least dimension of a compact homogeneous space through which a lattice action can factor. Assume that $\Gamma$ acts smoothly on a compact manifold $M$ preserving a volume form. Zimmer's volume-preserving conjecture. If $\dim(M)<d$, the action is isometric; if additionally $\dim(M)<n$ or $\Gamma$ is non-uniform, the action is finite. The conjecture is the volume-preserving form attributed to Zimmer and remains open in this generality.