Characterization of connected graphs whose geodesic metric satisfies the Menger condition

Let $G$ be a nonempty connected graph. Write $V(G)$ for its vertex set and $d_G$ for its geodesic distance. Let $\mathfrak{MB}$ denote the class of metric spaces $(X,d)$ such that

$$d(x,z)=d(x,y)+d(y,z)$$

whenever $d(x,z)\geqslant\max\{d(x,y),d(y,z)\}$. The graph characterization conjecture. The metric space $(V(G),d_G)$ belongs to $\mathfrak{MB}$ if and only if one of the following holds: $G$ is isomorphic to a path; $G$ is isomorphic to the cycle $C_4$; $G$ is isomorphic to a ray $R$; or $G$ is isomorphic to a double ray $DR$. This conjecture reformulates Menger's characterization in graph-theoretic language; the claimed classification connects the four-point metric condition with the global structure of connected graphs.