Regularity of the transition map between magnetic boundary parametrizations

Determine the regularity of the map $g_p^{-1}\circ g_q$ between the unit tangent circles at two points $p,q$ on the universal cover, where $g_p(v)=\gamma_v(\infty)$ parametrizes the forward magnetic boundary by the initial direction of the unit-speed magnetic geodesic $\gamma_v$.

Background

For a magnetic flow satisfying nonpositive magnetic curvature, the paper constructs a magnetic boundary at infinity and proves that the boundary parametrizations based at different points are homeomorphisms. In the uniformly hyperbolic case, the authors note that the corresponding transition map is nearly $C^2$. They leave unresolved what regularity follows under the weaker nonpositive magnetic-curvature assumption.