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

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

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

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

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

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