Zimmer's generalized low-dimensional action conjecture

Let $G$ be a semisimple Lie group with property $(T)$ and let $\Gamma$ be a lattice in $G$. Define $d$ as the least dimension of an infinite-image linear representation of $\Gamma$, define $n$ as the least dimension of a homogeneous space $K/C$ on which $\Gamma$ can act through a compact group, and set $b=\min\{n,d\}$. Zimmer's generalized low-dimensional action conjecture. If $M$ is a manifold with $\dim(M)<b-1$, every $\Gamma$ action on $M$ should be trivial. This is a broad form of Zimmer's conjecture, including lattices in groups such as $Sp(1,n)$ and $F_4^{-20}$; it remains open.