Conjecture on skew Anosov flows and regulating Reeb flows

Let $(\xi^+,\xi^-)$ be a bicontact structure supporting an Anosov flow $X$. A Reeb flow is bitransverse when it has the bitransversality property required in the paper, and a flow is regulating for a foliation when it regulates that foliation. Skew-and-regulating conjecture. The flow $X$ is positively skew if and only if:

• there exists a bitransverse Reeb flow $R^+$ of $\xi^+$ that is Anosov; and
• there exists a bitransverse Reeb flow $R^-$ of $\xi^-$ that is regulating for both the stable and unstable foliations of $X$.

The conjecture is known for the geodesic flow and, using bicontact surgeries, for the Handel–Thurston Anosov flow. The general case remains open.