Let $\Gamma$ be a Cayley graph of a finitely generated group. We can define the *visual boundary* of $\Gamma$ with respect to some base vertex $b$, denoted $\partial \Gamma$, as the set of geodesic rays based at $b$, modulo finite Hausdorff distance (since we have a transitive group action, the basepoint is not important). One can topologize this in the natural way, but that is not important here. We have the following straightforward question.

Suppose $\Gamma$ is a Cayley graph of a one-ended finitely generated group. Does $\partial \Gamma$ contain at least three points?

This is clearly known in the case of $CAT(0)$ and hyperbolic groups, but I'm interested in the most general case. It's not even immediately clear to me that $\partial \Gamma$ is non-empty, for example.

Any help or references would be appreciated. As with most questions about f.g. groups, I suspect this is either trivial or wrong.