Asymptotic product formula conjecture for weighted binary-sum Hankel determinants

Let $\mathcal H(n,1)$ denote the Hankel determinant studied in the paper. For each fixed integer $k\geq 0$, consider the values with index $n=2^m+k$. Asymptotic product formula conjecture. There exist integers $m_0=m_0(k)\geq 0$ and $r=r(k)\geq 0$, a sign $\varepsilon_k\in\{\pm 1\}$, a polynomial $P_k(X)\in\mathbb{Q}[X]$, and a constant $c_k\in\mathbb{Q}$ such that, for all $m\geq m_0(k)$,

$$\mathcal H(2^m+k,1)=\varepsilon_k P_k(2^m)\left(c_k+\sum_{j=1}^{m-r}\frac{2^{j-1}}{2^j-1}\right)\prod_{j=1}^{m-r}(2^j-1)^{2^{m-r-j}}.$$

Moreover, the polynomial $P_k(2^m)$ and the shift $r(k)$ are determined by the binary expansion of $k$ and the sequence of branches encountered when repeatedly applying the determinant recurrence theorem. This extends the explicit formulas known for $k=-1,0,1$; the general form and its dependence on the binary expansion remain conjectural.