Poincaré's homoclinic density conjecture

Let $M$ be a compact surface, let $\operatorname{Diff}_\mu^r(M)$ denote the space of $C^r$ area-preserving diffeomorphisms of $M$, and let $W^s(p)$ and $W^u(p)$ be the stable and unstable manifolds of a hyperbolic periodic point $p$. A point in $W^s(p)\cap W^u(p)$ is a homoclinic point of $p$. A subset $R$ is residual if it contains a countable intersection of open dense subsets. Poincaré's homoclinic density conjecture. There exists a residual set $R\subset\operatorname{Diff}_\mu^r(M)$ such that, if $f\in R$ and $p$ is a hyperbolic periodic point of $f$, then the homoclinic points of $p$ are dense in both stable and unstable manifolds of $p$. Equivalently, for every segment $J$ in $W^s(p)$ or $W^u(p)$,

$$\overline{W^s(p)\cap W^u(p)\cap J}=J.$$

The conjecture is presented as one of Poincaré's two fundamental conjectures on generic area-preserving surface dynamics; the supplied text gives no resolution status.