Herman's global instability conjecture for the N-body problem

Consider the spatial $N$-body problem with $N\geq 3$. Fix the center of mass at the origin. For each energy level $e$, let $H_e^{-1}(0)$ denote the corresponding energy surface, and reparametrize the flow there by a smooth positive function so that collisions occur only at infinite time. The non-wandering set consists of points whose every neighborhood meets one of its nonzero-time iterates. Herman's global instability conjecture. For every $e$, the non-wandering set of the Hamiltonian flow of $H_e$ on $H_e^{-1}(0)$ is nowhere dense in $H_e^{-1}(0)$. This is one of Herman's conjectures concerning the oldest problem in dynamical systems. The source presents it as open and notes that even the completeness of the collision-regularized flow is unclear because of possible non-collision singularities.