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

Let $n$ be an even integer, possibly nonpositive, and let $j>0$. Set

$$v=(n,-1).$$

Let $t$ be either the sequence $(-1)v^{j-1}$ or the sequence $v^{j-1}n$; both have length $2j-1$. The map $C_{-1,n}(-1,-)$ is defined using the extension of $C_{m,n}$ to distinct integer values of $m,n$ and finite sequences with entries in $\{m,n\}$. Negative-alphabet maximal-orbit conjecture. The orbit of either such sequence $t$ under $C_{-1,n}(-1,-)$ has length $2^j$. This generalizes the positive-alphabet maximal-orbit conjecture to the case $m=-1$.