Periodicity conjecture for integer sequences with bounded affine recurrence

Let $\lambda$ be a real number with $|\lambda|<2$. Consider an integer sequence $(a_k)_{k\in\mathbb Z}$ satisfying

$$0\le a_{k-1}+\lambda a_k+a_{k+1}<1$$

for every $k\in\mathbb Z$. Periodicity conjecture. Every such sequence is periodic. This conjecture concerns the dynamics of a class of discontinuous piecewise affine maps; the paper proves the assertion for several values of $\lambda$, including $\lambda\in\{\frac{\pm1\pm\sqrt5}{2},\pm\sqrt2,\pm\sqrt3\}$, while the full range $|\lambda|<2$ remains open in the supplied source.