Oscillation conjecture for zero-energy Kepler–Heisenberg orbits

Let $H$ be the Kepler–Heisenberg Hamiltonian, and let $z(t)$ denote the vertical coordinate of a zero-energy orbit. An orbit avoids the $z$-axis if it never meets the set where the horizontal coordinates vanish. Oscillation conjecture. All zero-energy orbits avoiding the $z$-axis have $z(t)$ oscillatory, meaning that $z(t)$ has infinitely many zeros without being identically zero on any interval. This property would provide the consecutive zeros required for the self-similarity theorem. Numerical evidence supports the conjecture, but attempts to prove it using Sturm comparison methods have failed; the restriction excluding the $z$-axis is necessary because there are counterexamples on that axis.