The high-rank lattice action conjecture

Let $G$ be a connected semisimple Lie group with finite center and real rank at least $2$, and let $\Gamma$ be an irreducible lattice in $G$. High-rank lattice action conjecture. The group $\Gamma$ has no nontrivial orientation-preserving action on $\mathbb{R}$ and is not left orderable. If no simple factor of $G$ is locally isomorphic to $\mathop{\mathrm{SL}}\nolimits(2,\mathbb{R})$, then $\Gamma$ has no faithful action on $S^1$, and whenever $\Gamma$ acts on $S^1$, every orbit is finite. The conjecture is verified in several cases, including certain arithmetic and non-cocompact cases, but remains open in general.