Depth-one foliations, pseudo-Anosov flows and universal circles

In 3-dimension topology, the study of foliations, flows and \pi_1-actions on 1-manifolds are closely related. Given a 3-manifold M, one can construct a \pi_1(M)-action on a circle from either a taut foliation or a pseudo-Anosov flow in M by works of Thurston, Calegari-Dunfield and Fenley. When the foliation is depth-one and the pseudo-Anosov flow is transverse to the foliation, we show (with some additional assumptions) that the circle actions from both settings are topologically conjugate. Moreover, the two circles admit extra structures that are compatible in the most natural sense.