The strong-edge conjecture for graphs of dynamical systems

Let $F$ be a dynamical system with a compact global attractor. An edge of the graph of $F$ is called strong when it satisfies the source's strong-edge relation. Strong-edge conjecture. Every edge of the graph of $F$ is strong. If true, this would rule out weak edges in graphs of systems with compact global attractors; the source reports that no counterexample was found but gives no proof.