 Solution

Moser theorem says that if $M$ is a compact smooth manifold without boundary and ${\omega }_t$, $0\le t\le 1$, is a smooth family of symplectic forms with constant de Rham cohomology class, then there is a smooth isotopy ${\psi }_t$, with ${\psi }_0=\mathrm{id}$, such thatThus the endpoints are symplectomorphic. The hypothesis is a whole path of symplectic forms, not merely equality of the endpoint de Rham cohomology classes.