Universal heterodimensional-cycle obstruction conjecture for convexity

Let $(\Sigma,\eta)$ be a contact Hamiltonian manifold, where $\eta$ is a contact Hamiltonian structure on $\Sigma$. A heterodimensional cycle is a pair of heteroclinic trajectories between hyperbolic invariant sets, and it is positive-negative when it contains a retrograde heteroclinic from a negative hyperbolic invariant set to a positive hyperbolic invariant set. Universal heterodimensional-cycle obstruction conjecture. $C^2$-generically, such heteroclinic cycles are the universal obstruction to convexity. Equivalently, a $C^2$-generic contact Hamiltonian structure is either convex or has robust positive-negative heterodimensional cycles. The conjecture proposes that robust positive-negative heterodimensional cycles account for all generic failures of convexity; the paper indicates that such cycles provide a robust obstruction, while the claimed universality remains conjectural.