Unboundedness conjecture for the 2-adic valuations of $b_m(n)$

Let $b_m(n)$ be the sequence considered in the paper and let $m\in\mathbb{N}_{\geq2}$ be such that $m$ is not of the form $2^k-1$ for any $k\in\mathbb{N}_{+}$. Unbounded valuation conjecture. The sequence

$$\bigl(\nu_2(b_m(n))\bigr)_{n\in\mathbb{N}}$$

is unbounded. This is proposed as a generalization of the preceding valuation observations, with the excluded values $m=2^k-1$ forming a separate family. The paper gives no general proof.