Asymptotic periodicity conjecture for single-piece Inverse Treblecross states

Let $[a X b]$ be a single-piece Inverse Treblecross state, with $a$ and $b$ denoting the lengths on either side of the piece, and let $\mathcal{G}$ denote its Sprague–Grundy value. Asymptotic periodicity conjecture. For any fixed $a$, the SG sequence of $[a X b]$ with respect to $b$ is eventually periodic with period $10$. There exists a minimal preperiod $b_a$ such that, for all $b>b_a$,

$$\mathcal{G}([a X (b+5)])=\mathcal{G}([a X b])\oplus 1.$$

Moreover,

$$b_a=\begin{cases}5,&a\in\{1,4\};\\0,&\text{otherwise}.\end{cases}$$

This conjecture is motivated by computed single-piece SG values and by the isolated anomalies identified for the exceptional residue classes; the asserted eventual periodicity and exact preperiod bound remain open.