Polynomial-time recognition of relating edges in graphs with no 6-cycles

Let $G$ be a graph in $\mathcal{G}(\widehat{C}_{6})$, and let $xy \in E(G)$ be an edge. The edge $xy$ is called relating when it has the relating-edge property used in the paper.

Relating-edge recognition conjecture. The following problem is polynomially solvable: given a graph $G \in \mathcal{G}(\widehat{C}_{6})$ and an edge $xy \in E(G)$, determine whether $xy$ is relating.

The conjecture is proposed as a common generalization of the paper's polynomial results for related graph classes. Its resolution is not given in the supplied text.