The analytic perturbation conjecture for elliptic symplectic surface maps

Let $(S,\Omega)$ be an analytic closed symplectic surface, and let $f\in \operatorname{Diff}^{\omega}_{\Omega}(S)$ be an analytic symplectic diffeomorphism displaying an elliptic periodic point. An analytic conservative perturbation of $f$ is an analytic perturbation preserving the symplectic area. Analytic perturbation conjecture. For every analytic and closed symplectic surface $(S,\Omega)$ and every analytic and symplectic $f\in \operatorname{Diff}^{\omega}_{\Omega}(S)$ which displays an elliptic periodic point, there are analytic and conservative perturbations of $f$ with positive metric entropy. This is described as the analytic counterpart of the authors' result with Turaev; no resolution is supplied in the source context.