Fried–Ghys genus one Birkhoff section conjecture for Anosov flows

Let $\phi^t$ be a transitive Anosov flow with orientable stable and unstable foliations. A genus one Birkhoff section is a Birkhoff section whose underlying surface has genus one. Fried–Ghys conjecture. The flow $\phi^t$ admits a genus one Birkhoff section. This conjecture asks whether the simplest useful Birkhoff sections always exist in the orientable transitive Anosov setting; the paper proves it under additional hypotheses, but the general claim remains open.