Palis's conjecture for flows with singularities

Let a flow, equivalently a vector field, be given on a manifold. A flow is hyperbolic if its chain recurrent dynamics are hyperbolic; a singular cycle is a cycle involving singularities, and a heterodimensional cycle is a cycle associated with invariant objects of different indices. Palis's conjecture. In any dimension, every flow (vector field) can be $C^r$-approximated by a hyperbolic one or by one displaying a homoclinic tangency, a singular cycle, or a heterodimensional cycle. The source states that this conjecture was proven for the $C^1$ topology on three-dimensional manifolds by Arroyo and Rodriguez-Hertz; the general formulation remains open in the supplied text.