Cartwright's Picard-group conjecture for products of graphs

Let $G$ and $H$ be finite graphs, let $\Delta$ be a triangulation of their product, and let $g(G)$ and $g(H)$ denote their genera. Write $\operatorname{Pic}(G)$, $\operatorname{Pic}(H)$, and $\operatorname{Pic}(\Delta)$ for the corresponding tropical Picard groups.

Cartwright's Picard-group conjecture.

$$\operatorname{Pic}(\Delta) \cong \operatorname{Pic}(G) \times \operatorname{Pic}(H) \times \operatorname{\mathbb{Z}}^{g(G)g(H)}.$$

This conjecture gives the Picard group of a triangulated product of graphs in terms of the Picard groups of its factors and an additional free abelian factor. The paper resolves the conjecture by proving the stated decomposition.