The no-all-Möbius periodic-orbit suspension conjecture

Let $F$ be a $C^k$ vector field on $S^3$, with $k\geq 3$. Let $\Lambda$ be a compact connected invariant set and $S$ a cross-section for $\Lambda$ such that the first-return map $h:S\to S$ is continuous on $S\cap\Lambda$, has infinitely many periodic points there, and its dynamics on $S\cap\Lambda$ factors through a subshift of finite type. No-all-Möbius suspension conjecture. There is no such vector field for which every periodic orbit $T\subseteq\Lambda$ satisfies $i(T)=0$. The conjecture is motivated by orientation constraints on flows on $S^3$ and the expectation that a symbolic hyperbolic-type invariant set cannot have all suspension periodic orbits of index zero; the source states that it is unresolved.