Density conjecture for generic orbits of the bicycle map

Fix a degree $d$ and consider the torus $\mathcal H_d$ of hedgehogs whose support functions are trigonometric polynomials of degree $d$, with the amplitudes of each harmonic fixed. The map $\mathcal M_\alpha$ acts on the $k$th circle factor by rotation through $2(\beta_k-\alpha)$, where $\tan\beta_k=k\tan\alpha$. Density conjecture. For a generic $\alpha$, the orbit of a point is dense in the torus $\mathcal H_d$. This expectation is based on the anticipated rational independence of the angles $\beta_k$; the supplied text gives no proof or resolution.