On $C^2$-stable effects of intermingled basins of attractors in classes of boundary-preserving maps

In the spaces of boundary-preserving maps of an annulus and a thickened torus, we construct open sets in which every map has intermingled basins of attraction, as predicted by I. Kan.
Namely, the attraction basins of each of the boundary components are everywhere dense in the phase space for such maps. Moreover, the Hausdorff dimension of the set of points that are not attracted by either of the components proves to be less than the dimension of the phase space itself, which strengthens the result following from the argument due to Bonatti, Diaz, and Viana.