Left orderable linear groups

Are all torsion-free finitely generated linear groups over $\mathbb{C}$ left orderable? In particular, are torsion-free congruence subgroups of $SL_n(\mathbb{Z})$ left orderable?

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The answer is no for congruence subgroups of $SL(n,\mathbb{Z})$ for $n \geq 3$. This is a theorem of Dave Witte-Morris; see