The periodicity and unboundedness conjecture for shifted Hankel determinant sequences

Let $n$ be a positive integer and let $\Delta_j^{(\ell)}$ denote the $j$th term of the Hankel determinant sequence shifted by $\ell$. Consider shifts with $\ell\geq n+2$.

The case $\ell\geq n+2$. The sequence $\Delta_j^{(n+2)}$ is $2n(n+1)$-periodic when $n$ is even and $2n(n+1)$-antiperiodic, hence $4n(n+1)$-periodic, when $n$ is odd, and its terms belong only to $\{-2,-1,0,1,2\}$, except when $n=1$, when they belong only to $\{-1,0,1\}$. All sequences $\Delta_j^{(\ell)}$ with $\ell\geq n+3$ are unbounded.

The statement extends the known periodicity results for shifts through $\ell=n+1$. It is proved for part (1) when $n=1$, while the general case remains open.