Diagonal uniqueness conjecture for Sprague–Grundy values of F-Wythoff

Let $\mathcal{G}(x,y)$ be the Sprague–Grundy value of the position $(x,y)$ in $\mathcal{F}$-Wythoff. For nonnegative integers $a$ and $g$, consider the diagonal parallel to the main diagonal given by positions $(b,a+b)$.

Diagonal uniqueness conjecture. For all nonnegative integers $a$ and $g$, there exists a unique integer $b$ such that

$$\mathcal{G}(b,a+b)=g.$$

For ordinary Wythoff's game, every such diagonal contains every nonnegative Sprague–Grundy value. The conjecture asserts the corresponding existence and uniqueness for $\mathcal{F}$-Wythoff; the source presents it as a conjecture based on computer explorations.