Hexality conjecture for bicontact forms on the surgered 3-torus

Let $T^3=\boldsymbol{R}^3/\boldsymbol{Z}^3$ and suppose $n_x>0$, $n_y<0$, and $n_z>0$. Let $M_{n_x,n_y,n_z}$ be the result of the specified surgeries on the coordinate curves, with Anosov flows $\phi_{xy}$ and $\phi_{yz}$ obtained from the two surgery descriptions. A bicontact Reeb-flow conjecture. There exists a bicontact form $(\alpha_+,\alpha_-)$ supporting $\phi_{xy}$ such that the Reeb flow $R_+$ of $\alpha_+$ is isotopically equivalent to $\phi_{yz}$. This is the first proposed instance of the paper’s hexality picture, in which the two Anosov flows exchange roles with positive Reeb flows; it remains open.