Equality of chain-limit points for a flow and its time-T map

Prove that for any continuous-time dynamical system F with a compact global attractor and its time-T map f=F^T (T>0), the set of chain-limit points for F coincides with the set of chain-limit points for f, where a point y is a chain-limit point of x if there exist ε-chains from x to y whose total durations T_ε tend to infinity as ε→0.

Background

The paper defines the downstream relation via ε-chains and distinguishes two notions of downstream approach: limit chain points (reachable via ε-chains whose durations diverge as ε→0) and trajectory chain points (reachable via ε-chains with uniformly bounded durations as ε→0). Earlier results (Proposition 4) show that a continuous-time system and its time-1 map have the same chain-recurrent sets, nodes, and edges.

The conjecture strengthens this by asserting equality of the finer structure of chain-limit points between a continuous-time system F and its discrete-time time-T map f=F^T.