Open sets of monotone ballistic trajectories in ABC flows

Consider initial conditions for the ABC flow in the non-KAM region. Principal-vortex conjecture. There exists an open set of positive measure of initial conditions such that $y$ (or $x$) grows monotonically and $x$ (or $y$) and $z$ are bounded and quasi-periodic. There also exists such an open set such that $x$ and $y$ grow monotonically and $z$ is bounded. Numerical experiments support the existence of these open sets; the source relates the claim to principal vortices but gives no proof or resolution.