Unboundedness of the coefficient sequence $t_m(n)$

Let $t_m(n)$ denote the coefficient sequence associated with the power series in the paper, and let $m\in\mathbb{N}_{\geq 2}$. Unboundedness conjecture.

$$\limsup_{n\to +\infty}t_m(n)=+\infty\quad\text{and}\quad\liminf_{n\to +\infty}t_m(n)=-\infty.$$

The preceding estimate gives only polynomial growth, while this conjecture asserts that every sequence with $m\geq2$ is unbounded in both directions. The paper does not establish the claim in general.