The three-edge balanced-circle conjecture for abelian gain graphs

Let $(\Gamma,g,G)$ be an abelian gain graph, where $G$ is a torsion-free abelian group, possibly $2$-divisible. Let $Z_1(\Gamma)$ be the integral cycle group, and suppose it has a basis consisting of balanced circles with at most $3$ edges. Three-edge balanced-circle conjecture. Then $(\Gamma,g,G)$ is balanced. The conjecture addresses when a basis of short balanced circles forces an entire abelian gain graph to be balanced. The corresponding four-edge assertion is false, while the source presents the three-edge case as unresolved.