General-region period-two scale invariance conjecture for CIS-Nim

Let $F$ be a forbidden set defining an instance of CIS-Nim, and let $S ameq\subseteq\mathbb{R}^3$ be open. For each $k$, let $\pi(S,k)$ denote the number of $P$-positions of the form $\{x2^k,y2^k,z2^k\}$, where $x,y,z\in\mathbb{Q}$ and $(x,y,z)\in S$. General-region period-two scale invariance conjecture. The limit

$$\lim_{k\rightarrow\infty}\frac{\pi(S,k)}{4^k}$$

converges. This proposes that the period-two scale invariance already proved for bounded regions extends to arbitrary open regions, though the source does not report a resolution.