The half-monodromy rotation-angle conjecture for nonplanar bicycling geodesics

Let $({\bf x}(t),{\bf y}(t))$ be a nonplanar bicycling geodesic with monodromy having rotation angle $\Delta\theta\in[0,2\pi)$ and translation $\Delta z>0$ about its axis. Let $I$ be the isometry associated with the geodesic, whose translation is $\Delta z/2$ and whose rotation angle is either $\Delta\theta/2$ or $\Delta\theta/2+\pi$. Rotation-angle conjecture. The rotation angle of $I$ is

$$\Delta\theta/2+\pi.$$

This conjecture is based on numerical evidence and concerns the unresolved choice between the two possible rotation angles for the isometry $I$ associated with a nonplanar bicycling geodesic.