Occurrence of an open family of closed orbits

Determine whether the union of closed orbits of the bi-rotational self-shrinker profile system can contain a nonempty open subset of the phase space, rather than being a measure-zero set.

Background

The recurrence theorem shows that for almost every phase point the trajectory is either closed or generates a complete nonproper self-shrinker with two non-escaping ends. The paper further proves that the union of closed orbits is either measure zero or contains a nonempty open set. It does not determine whether the second possibility actually occurs; if it does not, almost every initial condition yields the noncompact recurrent alternative.