The Fibonacci numbers $F_n$ can be given by $$\sum_{n\geq0}F_nx^n=\frac{x}{1-x-x^2}.$$ Among many many properties of this sequence, consider the following two results:

(1) the coefficients of the infinite product $\prod_{n=2}^{\infty}(1-x^{F_n})$ are $-1, 0$ or $+1$;

(2) the coefficients of the finite product $\prod_{n=2}^N(1-x^{F_n})$ are $-1, 0$ or $+1$.

It is almost irresistible to look for more. To this end, consider for example, the tribonacci sequences given by $$\sum_{n\geq0}T_nx^n=\frac{x^2}{1-x-x^2-x^3}.$$

QUESTION. Are the coefficients of the finite or infinite products $$\prod_{n=3}^N(1-x^{T_n}) \qquad \text{and} \qquad \prod_{n=3}^{\infty}(1-x^{T_n})$$ bounded? What is the best bound?

Remark. An affirmative answer lends itself further generalization of this assessment.