Labeled-pile and arbitrary-heap period-two scale invariance conjecture for CIS-Nim

Let $m$ be any positive integer, let $F$ be a set of positions in $m$-heap Nim with labeled piles, and let $S\subseteq\mathbb{R}^m$ be open. For each $k$, let $\pi(S,k)$ denote the number of $P$-positions of $m$-heap $\mathit{Nim}-F$ of the form $2^k v$ with $v\in S$. Labeled-pile and arbitrary-heap period-two scale invariance conjecture. The limit

$$\lim_{k\rightarrow\infty}\frac{\pi(S,k)}{2^{(m-1)k}}$$

converges. This extends the proposed scale invariance from three unlabeled piles to labeled piles and an arbitrary positive number of heaps; the source gives no evidence that it has been resolved.