Let's consider the $1$-variable rational function
$$F(z):=\frac{1-z}{(z^3 - z^2 + 2z - 1)\,(z^3 + z^2 + z - 1)}.$$

Numerical evidence convinces me of the truth of the following.
>**QUESTION.** Can you prove that $F(z)$ is positive, in the sense that its Taylor series at $z=0$ has positive coefficients?

**Note.** I'm not sure whether the concept of *multi-sections* of a series is efficient for the present purpose. Nor do I think that looking at *asymptotic growth* of largest positive real roots is any more elegant.