The homogeneous Hamiltonian torus-action conjecture

Consider the cotangent bundle $T^*(\mathbb{R}^n/\mathbb{Z}^n)$ with its standard symplectic structure. A homogeneous Hamiltonian torus action is an action compatible with the cotangent-bundle homogeneity, and two such actions are symplectically equivalent when related by a symplectomorphism. Homogeneous Hamiltonian torus-action conjecture. Up to symplectic equivalence, the only homogeneous Hamiltonian torus action is the standard one

$$\Phi_t(x,\xi)=(x+t\xi,\xi).$$

The source explains that this would imply the preceding toric-integrability conjecture and states that both conjectures were subsequently proved by E. Lerman and N. Shirokova.