Common eventual period under the sequential join

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 octal games. Write $\mathcal{G}_s$ for the scoring-play Sprague–Grundy function, and let $+_{\ell}$ denote the disjunctive sum and $\rhd$ the sequential join.

Sequential-join period conjecture. If $\mathcal{G}_s(n+_{\ell}m)$ eventually has period $p$, then $\mathcal{G}_s(n\rhd m)$ also eventually has period $p$.

The conjecture claims that the sequential join preserves the eventual period already present under the disjunctive sum. The paper does not establish it.