Asymptotic visibility of the one-dimensional RPSSL network in the AABBB regime

Prove that the one-dimensional RPSSL heteroclinic network (comprising one-dimensional connections on coordinate planes) is asymptotically visible for parameter values where the AABBB omnicycle is fragmentarily asymptotically stable. Demonstrate both Lyapunov-visibility and quasi-visibility of the entire one-dimensional network in this regime.

Background

In Section 3.3.3, the authors consider parameters for which the AABBB omnicycle is fragmentarily asymptotically stable and observe trajectories that visit the entire one-dimensional RPSSL network.

In Section 4.2, they explicitly conjecture that the whole network is in fact asymptotically visible—stronger than fragmentary visibility—inviting a rigorous proof.