Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as the 3-dimensional Heisenberg group), endowed with a left-invariant Riemannian metric.

The following statement can be found in the literature:

Any finitely generated group quasi-isometric to $Nil$ is in fact virtually isomorphic to a uniform lattice in $Nil$.

Recall that two groups are virtually isomorphic if they can be obtained one from the other via a finite number of steps, where each step consists in taking a finite extension or extracting a finite-index subgroup (or viceversa). The fact that virtually isomorphic groups are quasi-isometric is elementary, and any uniform lattice is quasi-isometric to its ambient Lie group as a consequence of Milnor-Svarc Lemma.

For example, the statement above is attributed to Gromov, ``Asymptotic invariants of infinite groups'' (1993), in the paper

``Quasi-isometries preserve the geometric decomposition of Haken manifolds'', Kapovich and Leeb, Inventiones Mathematicae (1997) and it is also stated in

`` Quasi-isometric classification of graph manifolds groups'', Behrstock and Neumann, Duke Math. J. (2008).

I tried to reconstruct the proof of the quasi-isometric rigidity of $Nil$, but I am not completely sure what should be the correct attribution. By Gromov's polynomial growth Theorem, any f.g. group quasi-isometric to $Nil$ is virtually nilpotent, and it is a classical result by Malcev that every f.g. (torsion free) nilpotent group is a uniform lattice in some nilpotent Lie group. Therefore, every f.g. group quasi-isometric to $Nil$ is virtually isomorphic to a uniform lattice in a (maybe different) nilpotent Lie group. Finally, maybe using the work of Pansu in ``Croissance des boules et des géodésiques fermées dans les nilvariétés'' (1983) one could probably show that the $Nil$ is recognized among the class of all nilpotent Lie groups by its asymptotic cone. This should conclude the proof of the quasi-isometric rigidity of $Nil$.

My question is: Is there an established reference for the quasi-isometric rigidity of $Nil$? If not, did someone write somewhere the details (if correct!) for the tentative proof I outlined above? Or is there a shorter argument that I did not spot?