Ultimate periodicity of Sprague–Grundy values along diagonals

Let $a_1$ and $a_2$ be nonnegative integers, and let $\mathcal{G}(a,b)$ denote the Sprague–Grundy value of the position $(a,b)$ in the game under consideration. Diagonal ultimate-periodicity conjecture. The sequence

$$(\mathcal{G}(a_1+i,a_2+i))_{i \geq 0}$$

is ultimately periodic. The paper has established ultimate additive periodicity in each fixed row, but this conjecture concerns diagonals parallel to the main diagonal; its general validity is left open.