Bifurcation conjecture for zero-energy orbits meeting the Heisenberg $z$-axis

Let $J$ denote the dilational momentum and consider a zero-energy orbit meeting the $z$-axis. Bifurcation conjecture. If

$$|J|<\frac{1}{2\sqrt{\pi}},$$

then the orbit has self-similarity like a generic $H=0$ orbit; in particular, $z(t)$ oscillates and the orbit looks like a dilating figure eight. If

$$|J|>\frac{1}{2\sqrt{\pi}},$$

then the orbit has self-similarity like a Heisenberg geodesic; in particular, $z(t)$ is monotonic and the orbit looks like a dilating helix. Numerical investigations suggest two qualitatively different families bifurcating at the threshold $|J|=\frac{1}{2\sqrt{\pi}}$, but the supplied text gives no proof.