Monotonicity conjecture for absolute values of weighted binary-sum Hankel determinants

Let $(n_k)$ be the sequence defined earlier in the paper, and let $\mathcal H(n,1)$ be the associated Hankel determinant. Monotonicity conjecture. The sequence of absolute values $(|\mathcal H(n,1)|)_{n\geq 2}$ is strictly increasing on each interval $[n_k,2^{k+1}+1]$ and strictly decreasing on each interval $[2^{k+1}+1,n_{k+1}]$. In particular, for each $k\in\mathbb{N}$,

$$|\mathcal H(2^k+1,1)|=\max_{n\leq 2^k+1}|\mathcal H(n,1)|.$$

The claim is motivated by experimental computations showing long monotone subsequences and apparent local maxima and minima at the indicated indices. Its validity for all indices remains open.