The singular-hyperbolicity exclusion conjecture for heteroclinic knots

Let $F$ be a smooth vector field on $S^3$ generating a heteroclinic knot $H$, and suppose that $Ess(F)$ with respect to $H$ is finite, possibly empty. An invariant set $I\subseteq S^3\setminus H$ is singularly hyperbolic when $F$ is singularly hyperbolic on $I$ in the sense that the flow admits the specified partially hyperbolic, volume-expanding or volume-contracting structure there, with all fixed points hyperbolic. Singular-hyperbolicity exclusion conjecture. There is no invariant set $I\subseteq S^3\setminus H$ on which $F$ is singularly hyperbolic and such that $H\subseteq\overline{I}$. This conjecture is motivated by the expectation that finite essential dynamics should not support complex dynamics accumulating on the heteroclinic knot; its resolution is not given here.