Sign-exception density conjecture for $t_m(n)$

Let $t_m(n)$ be the coefficient sequence in the paper. For a positive integer $m\geq2$ and $j\in\{0,1\}$, define

$$\mathcal{A}_{m,j}=\{n\in\mathbb{N}: \operatorname{sgn} t_m(3n+j)\neq(-1)^j\}.$$

Sign-exception density conjecture. The sets $\mathcal{A}_{m,j}$ are infinite and have asymptotic density zero:

$$\lim_{n\to +\infty}\frac{\#(\mathcal{A}_{m,j}\cap\{0,1,\ldots,n-1\})}{n}=0.$$

The conjecture refines the observed predominant signs $t_m(3n)>0$ and $t_m(3n+1)<0$ by asserting infinitely many exceptions but density zero. The paper provides numerical motivation but no proof.