Extremal Lagrangian torus conjecture for ellipsoids

Let $0<a_1\leq a_2\leq\dots\leq a_n<\infty$, and let $E^{2n}(a_1,a_2,\dots,a_n)$ be the corresponding ellipsoid with standard symplectic form. Ellipsoid extremal-torus conjecture. Every extremal Lagrangian torus in $(E^{2n}(a_1,a_2,\dots,a_n),\omega_{\mathrm{std}})$ is entirely contained in the boundary $\partial E^{2n}(a_1,a_2,\dots,a_n)$. The source says that this is believed to hold for ellipsoids of all dimensions and notes that the four-dimensional case follows from the paper's results.