Bonatti's periodic-orbit conjecture for singular chain recurrent classes

Let $X$ be a generic vector field, and let $C(\sigma)$ be a non-trivial chain recurrent class containing a hyperbolic singularity $\sigma$. A chain recurrent class is non-trivial when it is not reduced to a periodic orbit or a singularity. Bonatti's periodic-orbit conjecture. If $C(\sigma)$ is a non-trivial chain recurrent class containing a hyperbolic singularity $\sigma$, then $C(\sigma)$ contains a hyperbolic periodic orbit. This conjecture addresses the periodic content of generic singular chain recurrent classes and is used in the paper to motivate conditions in the surrounding results; its status is not given.