Robustly non-convex hypersurfaces in contact manifolds

We construct the first examples of hypersurfaces in any contact manifold of dimension 5 and larger that cannot be $C^2$-approximated by convex hypersurfaces, contrasting sharply with the foundational results of Giroux in dimension $3$ and Honda-Huang in the $C^0$ case. The main technical step is the first construction of a dynamical blender in the contact setting.