Eventual period-two conjecture for impartial scoring subtraction games

Let $S$ be a finite subtraction set, and let $\mathcal{G}_s(n)$ denote the scoring value of the subtraction game at heap size $n$. Write

$$k=\max\{S\}.$$

Eventual period-two conjecture. There exists an integer $N$ such that

$$\mathcal{G}_s(n+2k)=\mathcal{G}_s(n)$$

for all $n\geq N$.

This conjecture asserts eventual periodicity with period $2k$ for every finite subtraction set. The surrounding discussion notes that subtraction-game functions are eventually periodic, but asks whether this specific period always occurs; the example with subtraction set $\{4,5\}$ motivates the need for such an eventual statement.