Menzin's conjecture for bicycle monodromy

Let $F$ be a front-wheel path whose trajectory is a simple closed curve bounding a region of area $A$. For a bicycle of length $\ell$, let

$$M_F:{\mathbb S}^1\to {\mathbb S}^1$$

be the bicycle monodromy, which maps an initial rear-wheel position to the corresponding terminal position after traversing $F$. Menzin's conjecture. If $A>\pi\ell^2$, then $M_F$ has an attracting fixed point. This conjecture concerns the transition between elliptic and hyperbolic bicycle behavior and gives a geometric criterion ensuring that all relevant rear-wheel trajectories asymptotically approach a limiting trajectory. The source presents it as a conjecture originating with A. L. Menzin in 1906; no resolution is supplied here.