Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a group quasi-isometric (in the sense of Gromov) to $G$, can we conclude $H$ is left orderable? If the answer is no in the general case, what about 3-manifold groups?

Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a torsion-free group quasi-isometric (in the sense of Gromov) to $G$, can we conclude $H$ is left orderable? If the answer is no in the general case, what about 3-manifold groups?