Period-equality conjecture for nim-sequences and zero-position sequences

Let $G$ be a subtraction game, let $\mathcal{G}(n)$ denote the nim-value of a pile of size $n$, and define the zero-position indicator sequence $\mathcal{V}(n)$ by

$$\mathcal{V}(n)= \begin{cases} 0,&\text{if }\mathcal{G}(n)=0;\\ 1,&\text{otherwise}. \end{cases}$$

Period-equality conjecture. For every subtraction game, the sequences $\{\mathcal{G}(n)\}$ and $\{\mathcal{V}(n)\}$ have the same period $p$. Since subtraction-game nim-sequences are periodic, the indicator sequence is periodic as well; the conjecture asserts that passing to zero versus nonzero nim-values does not reduce the period. Its general status is open.