The HZ-conjecture for Reeb pseudo-rotations

Let $M$ be the smooth boundary of a star-shaped domain in ${\mathbb R}^{2n}$. A Reeb pseudo-rotation is a Reeb flow on $M$ with finitely many prime closed orbits. HZ-conjecture. The Reeb flow on $M$ has exactly $n$ prime closed orbits whenever it is a Reeb pseudo-rotation.

This is the contact analogue of the Hofer–Zehnder conjecture. The claim is almost proved in general, with lens spaces being the only manifolds admitting Reeb pseudo-rotations according to the supplied status evidence.