Palis-type conjecture for convex hypersurfaces and robust heterodimensional cycles

Let $(Y,\xi)$ be a contact manifold, and consider the space of its closed oriented hypersurfaces with the $C^2$ topology. A hypersurface is positive-negative when its characteristic foliation contains a heterodimensional cycle $C$ whose indices satisfy

$$\operatorname{ind}(C_+)\leq n-1\qquad\text{and}\qquad \operatorname{ind}(C_-)\geq n.$$

Palis-type conjecture. The space of closed oriented hypersurfaces in a contact manifold $(Y,\xi)$ decomposes into two open and $C^2$-dense sets: the set of convex hypersurfaces and the set of hypersurfaces containing a robust positive-negative heterodimensional cycle $C$. The two sets are disjoint in the results described by the source, and each is stated there to be nonempty and $C^0$-dense in the relevant higher-dimensional setting; the proposed $C^2$-dense decomposition remains open.