Hankel determinants for the $(n+2)$-shifted $q$-metallic numbers

Let $Φ_n(q)$ denote the $q$-deformation (in the sense of Morier-Genoud & Ovsienko) of the metallic number $ϕ_n=\frac{n+\sqrt{n^2+4}}{2}$. This is a power series with integer coefficients. We prove that the Hankel continued fraction of the $(n+2)$-shifted series $Φ_n^{(n+2)}$ is eventually periodic,