Conjecture on the unbounded coefficients of the companion series $C(q)$

Let $C(q)=\sum_{n\geq 0}c(n)q^n$ be the companion series whose coefficients are denoted by $c(n)$. Unboundedness conjecture.

$$\limsup_{n\to\infty}c(n)=+\infty,\qquad \liminf_{n\to\infty}c(n)=-\infty.$$

The conjecture asserts that the coefficients of $C(q)$ are unbounded in both the positive and negative directions. The paper gives numerical examples of coefficients outside $\{-2,-1,0,1,2\}$, but no resolution is provided.