Losert–Malcev conjecture on quasi-isometric nilpotent Lie groups

Let $G$ and $H$ be simply connected nilpotent Lie groups. Losert–Malcev conjecture. The groups $G$ and $H$ are quasi-isometric if and only if they are isomorphic. This would mean that the Losert–Malcev completion exactly captures the large-scale geometry of compactly generated locally compact groups with polynomial growth. The claim is presented as a conjecture and no resolution is given here.