Additive periodicity conjecture for R-Wythoff Sprague–Grundy sequences

Fix a nonnegative integer $a$, and let $\{\mathcal{G}_{\mathcal{R}}(a,i)\}_{i\geq 0}$ be the sequence of Sprague–Grundy values along the row with first coordinate $a$. A sequence $\{s_i\}_{i\geq i_0}$ is additively periodic if there exist integers $p\geq 1$ and $n_0\geq 0$ such that $s_{n+p}=s_n+p$ for every $n\geq n_0$. R-Wythoff additive periodicity conjecture. For every nonnegative integer $a$, the sequence $\{\mathcal{G}_{\mathcal{R}}(a,i)\}_{i\geq 0}$ is additively periodic. Additive periodicity is known for the corresponding row sequences in ordinary Wythoff's game, while the source presents this analogous behavior for R-Wythoff as a conjecture.