Abelianity conjecture for transitive graphs of non-negative Ollivier--Ricci curvature

Let $G$ be a transitive graph with non-negative Ollivier--Ricci curvature. Recall that two metric spaces are quasi-isometric if they are equivalent up to multiplicative and additive distortion, and let $\mathbb{Z}^d$ denote the rank-$d$ integer lattice. Abelianity conjecture. Every transitive graph of non-negative Ollivier--Ricci curvature is quasi-isometric to $\mathbb{Z}^d$ for some $d\geq 0$. In particular, any group having a Cayley graph of non-negative Ollivier--Ricci curvature must be virtually abelian. This is proposed by analogy with the Cheeger--Gromoll splitting theorem for cocompact non-negatively Ricci-curved manifolds; versions for groups under stronger curvature notions are known, but the graph statement remains open.