Arithmetic periodicity conjecture for partition games excluding {1,2}

Let $\mathcal C$ be a ruleset for the partition game $\operatorname{CUT}$. A sequence is ultimately arithmetic-periodic if it agrees with an arithmetic-periodic sequence from some point onward.

Arithmetic periodicity conjecture. Every instance $\mathcal C$ of $\operatorname{CUT}$ for which $\{1,2\}\not\subset \mathcal C$ is ultimately arithmetic-periodic.

This conjecture is proposed as a first step toward understanding whether every game $\operatorname{G}(\mathcal C)$ with $\mathcal C\neq\{1,2\}$ has a Sprague–Grundy sequence that is either ultimately periodic or ultimately arithmetic-periodic. The source gives no resolution of the conjecture; the case $\mathcal C=\{1,2\}$ is singled out as exhibiting unexplained drop-out values.