The real-rank conjecture for left orderability of arithmetic groups

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, and let $G_{\mathord{\mathbb{Z}}}$ denote its integral points. A subgroup is left orderable if it admits a total order preserved by left multiplication. The real-rank conjecture. If

$$\mathop{\operatorname{rank}}_{\mathbb{R}}G\geq 2,$$

then no finite-index subgroup $\Gamma$ of $G_{\mathord{\mathbb{Z}}}$ is left orderable. This extends the known result with the stronger hypothesis $\mathop{\operatorname{rank}}_{\mathbb{Q}}G\geq 2$; the conjecture concerns whether real rank can replace rational rank.