The higher-dimensional star-shaped Reeb-orbit two-or-infinity conjecture

Let $\Sigma^{2n-1}$ bound a star-shaped domain in $\mathbb{R}^{2n}$. Higher-dimensional two-or-infinity conjecture. For each such $\Sigma$,

$$\#\mathcal{T}(\Sigma)\in\{n\}\cup\{\infty\}.$$

The dimension-four case is solved by the Hofer–Wysocki–Zehnder result and its later confirmation, but the analogous conjecture for $n\geq 3$ is still widely open.