The maximal-orbit conjecture for $C_{1,n}$

Let $m\ne n$, let $s$ be a finite sequence, and let $C_{m,n}(s,-)$ be the length-preserving bijection on finite sequences over $\{m,n\}$. For even $n>0$ and $j>0$, let

$$t=1^{2j-1}.$$

Maximal-orbit conjecture for $C_{1,n}$. The orbit of $t$ under the map $C_{1,n}(1,-)$ has length $2^j$. The general orbit bound proved in the source shows that this is the maximum possible length for sequences of length $2j-1$, and the equivalent odd-length formulation concerns $E_{1,n}\left(1^{2^j},1^{2j}\right)$.