Extremal Lagrangian torus conjecture for convex toric domains

Let $X^{2n}_{\Omega}\subset \mathbb{C}^n$ be a convex toric domain, with standard symplectic form $\omega_{\mathrm{std}}$. Let $\operatorname{diagonal}(X^{2n}_{\Omega})$ denote the diagonal of the domain. Extremal toric-domain conjecture. Every Lagrangian torus in $(X_{\Omega},\omega_{\mathrm{std}})$ of symplectic energy equal to $\operatorname{diagonal}(X^{2n}_{\Omega})$ lies on the boundary $\partial X^{2n}_{\Omega}$. This extends the unit-ball conjecture to convex toric domains; the source states it as a conjecture, with the ellipsoid case discussed separately.