Geodetic circles generate the homology of locally finite graphs

Let $G$ be a locally finite graph. Let ${\mathcal G}_{\ell}(G)$ denote the set of homology classes represented by geodetic circles, and let ${\mathcal H}_{\ell}(G)$ denote the corresponding homology group. Write $\langle{\mathcal G}_{\ell}(G)\rangle$ for the subgroup or span generated by these classes.

Geodetic-generation conjecture. One has

$$\langle{\mathcal G}_{\ell}(G)\rangle={\mathcal H}_{\ell}(G).$$

This is the graph-theoretic generation claim underlying the continuum version discussed earlier. Its resolution is not supplied in the given text.