Non-geodesic isotopy classes of closed orbits

Establish that every possibly drilled pseudo-Anosov flow on a hyperbolic three-manifold admits closed orbits that are not ambiently isotopic to the unique geodesic in their respective free homotopy classes.

Background

Computations show that, for geometric veering triangulations with at most 10 tetrahedra, one can find a closed orbit that is not ambiently isotopic to the freely homotopic geodesic. The conjecture extends this computational phenomenon to all possibly drilled pseudo-Anosov flows on hyperbolic three-manifolds.