Eventual periodicity for finite taking-no-breaking octal games under the disjunctive sum

Let $O=(n_1n_2\dots n_t,p_1p_2\dots p_t)$ and $P=(m_1m_2\dots m_l,q_1q_2\dots q_l)$ be two finite taking-no-breaking octal games. Assume that at least one $n_s$ is neither $0$ nor $1$, and that whenever $n_i$ and $m_j$ equal $1$, $2$, or $3$, the corresponding scores satisfy $p_i=i$ and $q_j=j$, while otherwise $p_i=q_j=0$. Let $k$ be the largest entry in $O$ such that $n_k\neq 0,1$.

Disjunctive-sum periodicity conjecture. For every $m$, there exists $N$ such that

$$\mathcal{G}_s(n+2k+_{\ell}m)=\mathcal{G}_s(n+_{\ell}m)$$

for all $n\geq N$.

The conjecture was previously stated with substantial supporting evidence, and the paper studies whether the same periodic behavior persists for the other operators.