Compact-by-Lie conjecture for groups quasi-isometric to negatively curved homogeneous manifolds

A locally compact group is compact-by-Lie if it has a compact normal subgroup such that the quotient is a Lie group. Let $G$ be a compactly generated locally compact group, and let $X$ be a negatively curved homogeneous Riemannian manifold. Compact-by-Lie conjecture. If $G$ is quasi-isometric to $X$, then $G$ is compact-by-Lie. The statement is presented as a conjectural structural consequence of being quasi-isometric to a negatively curved homogeneous manifold; the source does not provide a resolution.