The conjecture on an open set of helical asymptotic orbits

Consider the Heisenberg-group dynamical system described above, with initial conditions and orbits under its Hamiltonian flow. An orbit is asymptotic to a helix if it approaches a helical trajectory in the relevant asymptotic sense. Helical-orbit conjecture. There is an open set of initial conditions whose orbits are asymptotic to helices.

This conjecture proposes that helical asymptotic behavior is not exceptional but occurs for a stable family of initial conditions. The preceding invariant-submanifold analysis exhibits conic behavior on a codimension-three submanifold, but does not establish the conjectured open-set behavior in the full dynamics.