Density of periodic points for Moser generic diffeomorphisms

Let $f:S^2\to S^2$ be an area-preserving diffeomorphism that is Moser generic. Write $\operatorname{Per}(f)$ for the set of periodic points of $f$. Density conjecture. The set of periodic points is dense in $S^2$:

$$\overline{\operatorname{Per}(f)}=S^2.$$

This is presented as the most fundamental open question concerning Moser generic diffeomorphisms and would describe the global abundance of periodic dynamics in the area-preserving setting. The source gives no resolution of the conjecture.