The conjecture that quasi-isometry to a nonamenable Cayley graph prevents the Liouville property

Let $G$ be a graph that is quasi-isometric to a nonamenable Cayley graph. The quasi-isometric non-Liouville conjecture. The graph $G$ is not Liouville. The source presents this as a question for general graphs and notes that the Liouville property is unstable under quasi-isometries; it gives no resolution.