Cornulier's quasi-isometric classification conjecture for solvable Lie groups

Let $S_1$ and $S_2$ be simply connected solvable Lie groups of real type, meaning that all eigenvalues of $\operatorname{ad}(X)$ are real for every $X$ in their Lie algebras. Cornulier's conjecture. The groups $S_1$ and $S_2$ are quasi-isometric if and only if they are isomorphic.

This conjecture would classify simply connected solvable Lie groups of real type up to quasi-isometry. The reduction to real-type groups uses the real shadow of a solvable Lie group; the conjecture is presented as an open problem concerning quasi-isometric classification.