The arithmetic-quotient fundamental-group conjecture for lattice actions

Let $G$ be a semisimple group whose simple factors have real rank at least $2$, let $\Gamma$ be a lattice in $G$, and let $\Gamma$ act faithfully and preserving volume on a compact manifold $M$. Assume the action is not isometric. The arithmetic-quotient fundamental-group conjecture. The group $\pi_1(M)$ should have a finite-index subgroup $\Lambda$ surjecting onto an arithmetic lattice in a Lie group $H$ such that $\operatorname{Aut}(H)$ locally contains $G$. This is an analogue for lattice actions of the fundamental-group conjecture for simple-group actions and remains open.