The non-lattice conjecture for SOL-like groups

Let $S$ be a SOL-like group. Suppose $\operatorname{Isom}(S,g)$ does not admit a uniform lattice for any left invariant Riemannian metric $g$ on $S$. The non-lattice conjecture. Then $S$ is not quasi-isometric to any finitely generated group.

This conjecture is proved when the constituent groups are Carnot or Carnot-by-Carnot with the relevant eigenspace-generated Lie algebras of dimension at least $2$. When one of these dimensions is $1$, the available conclusion is only that $S$ is not quasi-isometric to any amenable finitely generated group, so the full statement remains open in that case.