Akiyama–Brunotte–Pethő–Steiner periodic-orbit conjecture for digital rotations

Let $r_{\varphi}:{\mathbb Z}^2\mapsto{\mathbb Z}^2$ be the integer-part discretization of rotation by angle $\varphi$, defined by

$$r_{\varphi}(a,b)=\bigl(\lfloor a\cos\varphi-b\sin\varphi\rfloor,\lfloor a\sin\varphi+b\cos\varphi\rfloor\bigr).$$

Akiyama–Brunotte–Pethő–Steiner conjecture. Every orbit of $r_{\varphi}$ is periodic.

Computations and theoretical results support periodicity, and all orbits are known to be periodic for eleven specified values of $2\cos\varphi$. Infinitely many periodic orbits are known for every $\varphi$, but the assertion that every orbit is periodic remains unresolved.