The finite chain-recurrent-class conjecture away from homoclinic tangencies

Let $M$ be the underlying compact manifold, let ${\cal X}^1(M)$ denote the space of $C^1$ vector fields, and let $X\in{\cal X}^1(M)$. A vector field is approximable by homoclinic tangencies if it lies in the $C^1$ closure of vector fields with a homoclinic tangency. Finite-class conjecture. For $C^1$ generic $X\in{\cal X}^1(M)$ which cannot be approximated by vector fields with a homoclinic tangency, $X$ has only finitely many chain recurrent classes. This conjecture proposes finiteness of chain recurrent classes away from homoclinic tangencies and is stated without a resolution or known partial result in the supplied text.