The lattice left-orderability conjecture for lattices in special linear groups

Let $\Gamma$ be a lattice in $\mathop{\mathrm{SL}}\nolimits(n,\mathbb{R})$, with $n\geq 3$. A group is left orderable if it admits a left-invariant total order. Left-orderability conjecture. $\Gamma$ is not left orderable. This is the algebraic reformulation of the conjecture that such lattices have no faithful circle action, and the source states that the two formulations are equivalent; it is open.