Vertex-transitive graphs with uniformly bisecting quasi-geodesics

Suppose that $X$ is an infinite, connected, locally finite, quasi-transitive graph with the property that every bi-infinite quasi-geodesic uniformly coarsely separates $X$ into exactly two deep pieces. We show that such an $X$ is quasi-isometric to either the Euclidean plane or the hyperbolic plane. In particular, if $X$ is a Cayley graph of a finitely generated group $G$ with the above property, then $G$ is a virtual surface group. This can be interpreted as an extension of the well-known fact that a hyperbolic group with circular boundary is virtually Fuchsian. Our theorem positively resolves Problem 14.98 of the Kourovka Notebook, posed by V. A. Churkin in 1999. The proof uses an isoperimetric inequality of Varopoulos to show that if such a graph has the above property, then either it is hyperbolic or has quadratic growth.