Arbieto's conjecture on sectional-hyperbolic decomposition of generic star flows

Let $X$ be a $C^1$ flow. Its nonwandering set is the set of points $x$ such that every neighborhood $U$ of $x$ and every $T>0$ admit some $t\geq T$ with $X_t(U)\cap U\neq\emptyset$. A flow is star if it has a neighborhood in the space of $C^1$ flows such that every periodic orbit or singularity of every flow in that neighborhood is hyperbolic. A $C^1$ generic flow in a specified subset of the space of $C^1$ flows is one belonging to a residual subset there. A set is transitive when it has the transitivity property, and is positively or negatively sectional-hyperbolic in the sense used for the flow.

Arbieto's conjecture. The nonwandering set of a $C^1$ generic star flow is the disjoint union of finitely many transitive sets which are positively or negatively sectional-hyperbolic.

This conjecture predicts a finite sectional-hyperbolic decomposition of the nonwandering set for generic star flows. The supplied text lists it as one of two current conjectures relating star flows and sectional-hyperbolicity; no resolution is stated here.