Quasi-isometry implies commability for focal locally compact groups

Let $G$ be a focal locally compact group, and let $H$ be a locally compact group quasi-isometric to $G$.

Quasi-isometry–commability conjecture. The group $H$ is commable to $G$.

This would give a full quasi-isometric classification of focal locally compact groups in terms of commability. The surrounding results establish invariance of the invariant $\varpi$ under quasi-isometry and provide partial classification results, but the asserted implication remains unresolved.