Smooth vs. topological Anosov flows in the non-transitive case

Determine whether every non-transitive topological Anosov flow on a closed 3-manifold is orbit equivalent to a smooth Anosov flow. Equivalently, ascertain whether the notions of smooth Anosov flow and topological Anosov flow coincide up to orbit equivalence in the non-transitive setting.

Background

The appendix distinguishes between smooth ("true") Anosov flows and topological Anosov flows as considered in prior classification work. For transitive flows, these notions are known to coincide up to orbit equivalence by results of Shi (referenced as [Sh21]). However, the equivalence of these two notions in the non-transitive case has not been established.

This question is relevant to the broader program classifying partially hyperbolic diffeomorphisms via associated Anosov flows and self orbit equivalences, particularly because some arguments and constructions rely on smoothness while earlier frameworks allowed topological Anosov flows.