Ultimate arithmetic periodicity for Family C cut games

Let $\mathcal{C}$ be a cut set satisfying

$$\{1,2\}\subseteq\mathcal{C},\qquad 3\notin\mathcal{C},$$

and suppose that $\mathcal{C}\neq\{1,2\}$. This is Family C. Family C arithmetic-periodicity conjecture. The nim-sequence for every game of \textsc{cut} in Family C is ultimately arithmetic-periodic. The paper identifies Family C as one of the families whose patterns are not yet clear and records this observed ultimate periodicity as an open conjecture.