The star-shaped Reeb-orbit multiplicity conjecture

Let $\Sigma^{2n-1}$ be the boundary of a star-shaped domain in $\mathbb{R}^{2n}$, and let $\mathcal{T}(\Sigma)$ denote the set of prime closed Reeb orbits on $\Sigma$. Multiplicity conjecture. For each such $\Sigma$,

$$\#\mathcal{T}(\Sigma)\geq n.$$

This is a long-standing multiplicity conjecture. The paper summarizes proofs under pinching, index, non-degeneracy, or other additional assumptions, while the unrestricted statement remains open in general.