Arithmetic periodicity for Family A cut games

Let $X$ be a non-empty set of even numbers, each at least $4$, and let $Y$ be a non-empty set of odd numbers, each at least $5$. Write $x=2c$ for the smallest element of $X$. Family A consists of cut sets

$$\mathcal{C}=\{1,3\}\cup X$$

or

$$\mathcal{C}=\{1,3\}\cup X\cup Y.$$

Family A arithmetic-periodicity conjecture. The nim-sequence for every game of \textsc{cut} in this family is precisely $(0,1)^c (+2)$. This concerns one of the families whose nim-sequence was described as unknown; proving arithmetic periodicity for Families A and B would imply Conjecture 1 of the cited prior work.