The PYY conjecture on singular star flows and aperiodic classes

A singular star flow is a star flow with singularities, and a singular aperiodic class is a chain-recurrent class containing singularities but no periodic orbit. A homoclinic class is the closure of transverse intersections of the stable and unstable manifolds associated with a periodic orbit. The PYY conjecture. $C^1$-generic singular star flows do not admit singular aperiodic classes. Consequently, they have only finitely many chain-recurrent classes, all of which are homoclinic classes of periodic orbits. The source states that this is the remaining open half of the preceding star-flow conjecture; proving it would rule out singular aperiodic classes and yield the asserted finiteness and homoclinic-class decomposition.