Explicit numerator conjecture for the orbit of $\Psi_1$

Let

$$\Psi_1:=\left\langle\begin{pmatrix}1&1\\1&1\end{pmatrix},\begin{pmatrix}1&0\\1&1\end{pmatrix}\right\rangle^+,$$

and consider its orbit of the indicated vector. Explicit $\Psi_1$ numerator conjecture. Every non-square integer $n>10569$ appears as a numerator in the orbit $\Psi_1\begin{pmatrix}2\\2\end{pmatrix}$. This is presented as a computationally motivated explicit instance of the preceding local-global expectation; no proof or resolution is supplied.