The algebraic version of the irreducible arithmetic-group conjecture

Let $\Gamma$ be a group. A total order $\prec$ on $\Gamma$ is left-invariant if $a\prec b$ implies $ca\prec cb$ for all $a,b,c\in\Gamma$. The algebraic version of the conjecture. If $\Gamma$ is an irreducible arithmetic group, then $\Gamma$ does not have a left-invariant total order unless $\Gamma$ is an arithmetic subgroup of a very small Lie group. For countable groups, having a faithful action on $\mathbb{R}$ is equivalent to admitting a left-invariant total order, so this is presented as an algebraic restatement of the main conjecture. The source gives no evidence that this version has been resolved.