Preperiod conjecture for extensions of subtraction sets of size at most three

Let $A$ be a finite subtraction set with $|A|\leq 3$, let $S\in\{0,1\}^{\alpha}$ be a seed, and write $p=\operatorname{Per}(A,S)$. An integer of the form $kp+x$ is an extension of $A$ for every $k\in\mathbb{N}_+$ and $x\in A$ under the condition stated. Preperiod-extension conjecture. If $kp+x$ is an extension of $A$ for all $k\in\mathbb{N}_+$ and $x\in A$, then

$$\operatorname{PrePer}(A,S)=0.$$

The statement is the converse of the cited invariant-extension proposition for sets of size at most three. The source notes that it suffices to check $k=1$, and that the converse is not true for nonempty seeds in the unrestricted setting.