Weak covering conjecture for infinite Cayley graphs

Let $\Gamma$ be an infinite Cayley graph that is not quasi-isometric to $\mathbf Z$. A graph $\Delta$ is a cover of $\Gamma$ if there is a covering map from $\Gamma$ to $\Delta$; write $\Delta\not\simeq\Gamma$ when the graphs are not isomorphic. Weak covering conjecture. Every infinite Cayley graph $\Gamma$, not quasi-isometric to $\mathbf Z$, covers an infinite transitive graph $\Delta\not\simeq\Gamma$. This is a weak form of a question of Benjamini and Duminil-Copin concerning coverings of infinite transitive graphs by graphs of bounded girth. The proposed statement is disproved by possible counterexamples arising from Cayley graphs of strongly simple groups.