The S-integer conjecture for left orderability of arithmetic groups

Let $G$ be a ${\mathord{\mathbb{Q}}}$-simple algebraic ${\mathord{\mathbb{Q}}}$-group, let $\{p_1,\ldots,p_n\}$ be a nonempty set of prime numbers, and write $G_{\mathord{\mathbb{Z}}[1/p_1,\ldots,1/p_n]}$ for the corresponding group of $S$-integer points. A subgroup is left orderable if it admits a total order preserved by left multiplication. The S-integer conjecture. No finite-index subgroup $\Gamma$ of

$$G_{\mathord{\mathbb{Z}}[1/p_1,\ldots,1/p_n]}$$

is left orderable. This proposes an analogue over rings of $S$-integers without any restriction on the real rank; the paper proves related results for several nonarchimedean settings, but the stated general conjecture remains open.