Singular hyperbolicity and finiteness of chain recurrence classes for star flows

Let $X$ be a vector field and let its chain recurrent set be the set of points returning to themselves by arbitrarily small pseudo-orbits. A vector field is a star vector field if all its periodic orbits remain hyperbolic for every sufficiently $C^1$-nearby vector field; a chain recurrence class is an equivalence class under mutual chain attainability. Star-flow singular hyperbolicity conjecture. For every star vector field, the chain recurrent set is singular hyperbolic and consists of finitely many chain recurrent classes. The source says this conjecture appears in work of Gan, Wen, and others and was proven for an open and dense set of vector fields in dimension four; its full generality is left unresolved in the supplied text.