Boundedness conjecture for discretised rotations

Let

$$F:\mathbb{Z}^2\to\mathbb{Z}^2,\qquad (x,y)\mapsto(\lfloor\lambda x\rfloor-y,x),$$

where $\lambda$ is real and $|\lambda|<2$. The map is the discretisation of the linear rotation-like map $(x,y)\mapsto(\lambda x-y,x)$, with $\lambda=2\cos(2\pi\nu)$. Boundedness conjecture. For every real $\lambda$ with $|\lambda|<2$, all orbits of $F$ are periodic. This question concerns the near-integrable behaviour of discretised rotations and remains unsolved; a related general conjecture on boundedness was previously formulated for discretised Hamiltonian rotations.