Lazzarini's boundary conjecture for extremal Lagrangian tori

Let $D^{2n}$ be the $2n$-dimensional unit disc with the standard symplectic form $\omega_0$, and let $S^{2n-1}=\partial D^{2n}$. A Lagrangian torus $L\subset D^{2n}$ is extremal when it realizes the Lagrangian capacity of the disc. Lazzarini's boundary conjecture. Every extremal Lagrangian torus $L\subset (D^{2n},\omega_0)$ is contained in the boundary

$$\partial D^{2n}=S^{2n-1}.$$

The Clifford torus gives an extremal example contained in the boundary, and in dimension two it is the only extremal Lagrangian torus. The conjecture asks whether this boundary containment holds in every dimension.