The approximation conjecture for three-dimensional flows

A three-dimensional flow is a $C^1$ vector field on a three-dimensional manifold. A flow has a homoclinic tangency when its stable and unstable manifolds have a nontransverse intersection, and it is singular-Axiom A without cycles when its nonwandering set has a spectral decomposition and there are no finitely many nonsingular orbits joining the pieces of that decomposition in a cyclic way. Here, $C^1$ approximation means approximation in the $C^1$ topology.

Approximation conjecture. Every three-dimensional flow can be $C^1$ approximated by a flow exhibiting a homoclinic tangency or by a singular-Axiom A flow without cycles.

This is the conjecture attributed in the source to Morales and Pacifico and is presented as a problem concerning the possible dynamics of three-dimensional flows. The supplied text gives no resolution, so its status is left open.