Weinstein's conjecture for rotational Beltrami flows

Let $(M,g)$ be a Riemannian three-manifold. A rotational Beltrami flow is a smooth steady-state flow whose vector field is parallel to its curl; here smooth means $C^\infty$. A closed orbit is a periodic flowline.

Hydrodynamical Weinstein conjecture. Every smooth rotational Beltrami flow on a Riemannian three-manifold has a closed orbit.

Via the correspondence between nonsingular Beltrami fields and Reeb fields, this is the hydrodynamical formulation of the Weinstein conjecture. The paper derives the claim on $S^3$ from Hofer's theorem and presents the general statement as a conjecture.