The Hankel determinant periodicity and recurrence conjecture for q-deformed metallic numbers

Let $k$ be a positive integer, and for $ll\in\mathbb{Z}_{\geq 0}$ let

$$\Delta_n^{(\ell)}:=\Delta_n^{(\ell)}\bigl(\left[y_k\right]_q\bigr)$$

denote the $\ell$-shifted sequence of Hankel determinants associated with the $q$-deformation of the metallic number $y_k$.

Hankel determinant periodicity and recurrence conjecture. The $k+2$ sequences $\Delta_n^{(0)},\Delta_n^{(1)},\dots,\Delta_n^{(k+1)}$ take only the values $-1,0,1$. They are $2k(k+1)$-periodic when $k$ is even and $2k(k+1)$-antiperiodic, hence $4k(k+1)$-periodic, when $k$ is odd:

$$\Delta_{n+2k(k+1)}^{(\ell)}=(-1)^k\Delta_n^{(\ell)}\quad\text{for all }n\in\mathbb{Z}_{\geq 0}.$$

Moreover, all these sequences satisfy

$$\Delta_{n+2k+2}^{(\ell)}\Delta_n^{(\ell)}=\Delta_{n+2k+1}^{(\ell)}\Delta_{n+1}^{(\ell)}-\bigl(\Delta_{n+k+1}^{(\ell)}\bigr)^2\quad\text{for all }n\in\mathbb{Z}_{\geq 0}.$$

For $\ell=1,2,\ldots,k+1$, the consecutive shifted sequences are interconnected by

$$\Delta_n^{(\ell)}=(-1)^{n+\frac{k(k+2\ell+1)}{2}}\Delta_{n+k+1}^{(\ell-1)}\quad\text{for all }n\in\mathbb{Z}_{\geq 0}.$$

The conjecture arose from computer experimentation and concerns the simultaneous integrality, periodicity, Somos-type recurrence, and relations among shifted Hankel determinant sequences associated with $q$-deformed metallic numbers. The supplied context does not state whether it has been proved or disproved.