Positivity and monotonicity conjecture for the sequence f_{2^k,n}

Let $f_{2^k,n}$ denote the sequence of values defined in the paper, and let $k\geq 1$ and $\nu\geq 0$ be integers. For each interval of integers

$$\nu\,2^{k+1}\leq n\leq(\nu+1)\,2^{k+1}-2,$$

consider the signed sequence $((-1)^\nu f_{2^k,n})_n$. Positivity and monotonicity conjecture. On this interval, $(-1)^\nu f_{2^k,n}>0$, and the sequence is strictly increasing except that its final two terms are equal. Computations indicate that the proposition proved for the first interval extends to every interval between successive zeros.