Converse compatibility conjecture for connected tensor products of signed graphs

Let $\Sigma_1$ and $\Sigma_2$ be signed graphs, and let $\Sigma_1\otimes\Sigma_2$ denote their connected tensor product. A signed graph is compatible when every pair of its vertices joined by multiple shortest paths has the same signed distance along those paths.

Converse compatibility conjecture. If $\Sigma_1$ and $\Sigma_2$ are two compatible signed graphs, then the connected tensor product $\Sigma_1\otimes\Sigma_2$ is compatible.

The conjecture is presented as the converse of the preceding theorem, which establishes that compatibility of the connected tensor product implies compatibility of each factor. Its general resolution is not given in the source.