A non-plain virtually free group with a strictly 2-geodetic Cayley graph

We construct an explicit infinite virtually free group $G=C_4{\ast }_{C_2}{D}_3$ with a finite inverse-closed generating set whose Cayley graph is $2$-geodetic but not geodetic. More precisely, we describe the full language of geodesics and show that every element of $G$ has a unique geodesic representative except for a single element, which has exactly two. This gives a counterexample to the conjecture that an infinite group admitting a $k$-geodetic Cayley graph must admit a geodetic Cayley graph with respect to the same generating set. We further prove that $G$ is freely indecomposable and hence non-plain. Consequently, Shapiro’s conjecture that every geodetic group is plain and the conjecture that every hyperbolic $k$-geodetic group is geodetic cannot both hold.