Irie's strong closing conjecture for ellipsoid boundaries

Let $E \subset \mathbb{C}^n$ be an ellipsoid and let $(\partial E,\lambda|_{\partial E})$ be its contact boundary, where $\lambda$ is the standard Liouville form. Irie's strong closing conjecture. The boundary $(\partial E,\lambda|_{\partial E})$ has the strong closing property. This conjecture concerns the existence of strong closing phenomena for the Reeb dynamics of ellipsoid boundaries; the source presents it as a conjecture motivated by the difficulty of computing the relevant $U$-map.