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

Given a taut depth-one foliation $\mathcal{F}$ in a closed atoroidal 3-manifold $M$ transverse to a pseudo-Anosov flow $\phi$ without perfect fits, we show that the universal circle coming from leftmost sections ${\mathfrak{S}}_{\mathrm{l}\mathrm{e}\mathrm{f}\mathrm{t}}$ associated to $\mathcal{F}$, constructed by Thurston and Calegari-Dunfield, is isomorphic to the ideal boundary of the flow space associated to $\phi$ with natural structure maps. As a corollary, we use a theorem of Barthelmé-Frankel-Mann to show that there is at most one pseudo-Anosov flow without perfect fits transverse to $\mathcal{F}$ up to orbit equivalence.