Partial arithmetic-periodicity conjecture for Family B 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 $5$. Let $x$ and $y$ be the smallest elements of $X$ and $Y$, respectively. Family B consists of cut sets

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

Family B partial conjecture. If $3x<y$, then

$$\mathcal{G}_{\mathcal{C}}(n)=\mathcal{G}_{\{1,x\}}(n)$$

for $n\geq 1$. The paper presents this as a partial extension of the known result for two-element cut sets and notes that no full conjecture is available for Family B; proving arithmetic periodicity for Families A and B would imply Conjecture 1 of the cited prior work.