Generic shadowing and weak specification for non-Anosov area-preserving maps

Let $M$ be a compact surface, and consider the space of $C^1$ area-preserving maps of $M$ with its $C^1$ topology. A map is Anosov if it has a uniformly hyperbolic invariant splitting. The conjecture. For non-Anosov area-preserving maps, the set of maps that do not have the shadowing property or the weak specification property is meagre in the $C^1$ topology. This would extend the recent results mentioned in the source to the low-dimensional conservative setting; the conjecture is related to the unresolved robustness of homoclinic tangencies and to the $C^1$-density of hyperbolicity on surfaces.