The scalar strong BMV coefficient conjecture

Let $g\in\mathbf{P}^{\mathrm{pos}}$, let $a_i,b_i$ be positive, and let $n$ be a positive integer. Define polynomials $f_i(x)$ by

$$\prod_{i=1}^n\bigl(x+g(a_i+b_i y)\bigr)=\sum f_i(x)y^i.$$

The scalar strong BMV coefficient conjecture. Every $f_i$ belongs to $\mathbf{P}^{\mathrm{pos}}$, and consecutive coefficients satisfy

$$f_i\longleftarrow f_{i+1}.$$

The source presents this as a special case of the composition conjecture and does not prove it in general.