The two-or-infinitely-many periodic Reeb orbits conjecture

Let $(M,\alpha)$ be a closed connected $3$-manifold with a contact form. The associated Reeb flow has periodic orbits, and the claim concerns the number of such orbits.

Two-or-infinitely-many periodic Reeb orbits conjecture. The associated Reeb flow has either exactly two or infinitely many periodic orbits.

This conjecture was recently established for contact structures with torsion first Chern class and, without a torsion assumption, for non-degenerate contact forms. The general case remains open.