Hexality conjecture for swapped contact structures and Anosov flows

Suppose $n_x>0$, $n_y<0$, and $n_z>0$, and let $M_{n_x,n_y,n_z}$ carry the Anosov flows $\phi_{xy}$ and $\phi_{yz}$ from the two surgery descriptions. A contact-structure swapping conjecture. There exist two positive contact structures $\xi_x,\xi_z$ and a negative contact structure $\xi_y$ on $M_{n_x,n_y,n_z}$ such that $\phi_{xy}$ is isotopically equivalent to a Reeb flow of $\xi_z$ and is supported by $(\xi_x,\xi_y)$, while $\phi_{yz}$ is isotopically equivalent to a Reeb flow of $\xi_x$ and is supported by $(\xi_z,\xi_y)$. This combines the two symmetric Reeb-flow predictions and asserts that the Anosov flows and positive contact structures can be interchanged; it remains open.