Periodicity conjecture for weighted finite taking-no-breaking games

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 $m_j$ are in $\{1,2,3\}$, 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 $n_k\neq 0,1$. Taking-no-breaking periodicity conjecture. Then

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

Here $+_{\ell}$ denotes disjunctive sum, and the conjecture asserts that the period is $2k$ even after adding the fixed game $m$. The authors report strong computational evidence but say that proving the claim is surprisingly difficult; the conjecture is open in the supplied text.