Hofer-Wysocki-Zehnder's two-or-infinity conjecture

Let $Y$ be a star-shaped hypersurface in $\mathbb{R}^4$, and call a periodic orbit simple if it is not a multiple cover. Hofer-Wysocki-Zehnder's two-or-infinity conjecture. A Hamiltonian flow along any star-shaped hypersurface in $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits. The conjecture is presented as a prediction for low-dimensional Reeb dynamics and is later resolved affirmatively in the source by a theorem applying more generally to suitable contact flows on rational homology spheres.