Symplecto-rigidity of all Anosov symplectomorphisms

Prove that every Anosov symplectomorphism is symplecto-rigid, meaning that its defining symplectic form is the unique continuous invariant symplectic form in its cohomology class, or equivalently that it admits no nonzero continuous invariant exact 2-form.

Background

The paper defines an Anosov symplectomorphism as symplecto-rigid when its symplectic form is the unique continuous invariant symplectic form in its cohomology class. This is equivalent to the nonexistence of nontrivial continuous invariant exact 2-forms.

The authors establish symplecto-rigidity for several classes, including a residual set of Anosov symplectomorphisms and sufficiently small perturbations of certain four-dimensional symplectic toral automorphisms. They also note that the conjecture was previously proved for Anosov automorphisms of nilmanifolds, but the general case remains unresolved.