Positive rotation-number conjecture for convex bicycle tracks

Let $\Gamma$ be a simple strictly convex curve with hyperbolic or parabolic monodromy, and let $\gamma$ be the corresponding rear track with rotation number $\rho(\gamma)$. Positive rotation-number conjecture. One has

$$\rho(\gamma)=1.$$

The source presents this as one of two conjectures based on computer experiments. It notes that the claim would imply the stated length bound and the equality characterization in the parabolic case, but provides no proof.