All edges in the chain graph are strong

Prove that for any dynamical system F with a compact global attractor, every directed edge A→B between distinct nodes (maximal chain-recurrent sets) in the chain graph is strong; that is, there exists a two-sided trajectory τ with nonempty backward and forward limit sets satisfying α(τ)⊂A and ω(τ)⊂B.

Background

In the chain graph, edges A→B are defined by the downstream relation, while a strong edge additionally requires the existence of a two-sided trajectory whose backward limit set lies in A and forward limit set lies in B. The authors have not found a counterexample with a weak (non-strong) edge and conjecture that all edges are strong when a compact global attractor exists.