Conjecture on periodicity of complementary Beatty-sequence games

Let $(a_n)=(\lfloor\alpha n+\delta\rfloor)$ and $(b_n)=(\lfloor\beta n+\gamma\rfloor)$ be complementary Beatty sequences, with $\alpha,\beta,\delta,\gamma\in\mathbb{R}$ and $\alpha,\beta>0$. Let $G$ be the invariant game whose move set is

$$\mathcal{M}(G)=\{(a_n,b_n)\mid n\in\mathbb{N}\},$$

with symmetric moves also allowed. A set of positions is periodic when every sufficiently large position in it lies on an infinite arithmetic ray as defined in the source.

Beatty-periodicity conjecture. The set $\mathcal{P}(G)$ is periodic if and only if the modulus $\alpha$ of $(a_n)$ is rational. This proposes a complete rational-versus-irrational criterion for periodicity of the $P$-positions of these invariant subtraction games; the source leaves it open.