The ordered-chaotic-mixed classification conjecture for invariant one-dimensional sets

Let $F$ be a smooth vector field on $S^3$ with fixed points $\{x_1,\ldots,x_n\}$, and let $H$ be a one-dimensional flow-invariant set such that each component $C$ of $H\setminus\{x_1,\ldots,x_n\}$ satisfies one of the stated alternatives: $C$ is homeomorphic to $S^1$, $C$ is homeomorphic to $(0,1)$, or every fixed point lies in the closure of some such component. For a periodic orbit $T$ of $F$, write $i(T)$ for its Orbit Index. Ordered-chaotic-mixed classification conjecture. Under these assumptions, $F$ can be deformed relative to $H$ to a smooth vector field $G$ satisfying precisely one of the following: $G$ is ordered, generating finitely many attracting or repelling periodic orbits $T_1,\ldots,T_n$ with $i(T_j)=1$ and $Ess(F)$ equal to $\{T_1,\ldots,T_n\}$; $G$ is chaotic, generating infinitely many periodic orbits, all with Orbit Indices $-1$ or $0$, persisting under deformations relative to $H$ without changing knot type or collapsing together, so that $Ess(F)$ is infinite; or $G$ is mixed, with an invariant surface $S$ satisfying $S\cap H\subseteq\{x_1,\ldots,x_n\}$ such that the flow on every component of $S^3\setminus(H\cup S)$ is ordered or chaotic. This conjecture proposes a Thurston–Nielsen-type classification of the dynamics obtainable after deformation relative to the invariant set; the source provides examples of finite and infinite essential classes but no proof of the classification.