Infinitely many bi-normal trajectories near the primaries in the spatial circular restricted three-body problem

Let $q(t)=(q_1(t),q_2(t),q_3(t))$ be a trajectory in the spatial Circular Restricted Three-Body Problem, in the low-energy range and near the primaries. A trajectory is bi-normal to the $x$-axis when there exist times $t_0<t_1$ such that

$$q_2(t_j)=q_3(t_j)=\dot{q}_1(t_j)=0$$

for $j=0,1$, with the initial velocity pointing strictly outward from the Earth–Moon axis. Bi-normal trajectory conjecture. Assuming the twist condition, or a variation thereof, there exist infinitely many trajectories bi-normal to the $x$-axis in the low-energy range and near the primaries. Such a trajectory starts on the Earth–Moon axis with velocity pointing strictly outward and returns after finite time to satisfy the same condition. This would extend the Poincaré–Birkhoff-type result to chords on the boundary of the relevant Liouville domain; the conjecture remains open in the supplied source.