Mahler-type functional equation for generating functions of polynomial powers

Let $f(x)$ be a polynomial over a finite field, let $p$ be the relevant prime, and let $g_{f(x),p}(z)$ denote the generating function associated with the coefficient sequence for powers of $f(x)$. Functional-equation conjecture. For any $f(x)$ and $p$, the generating function $g_{f(x),p}(z)$ satisfies

$$r(z)g_{f(x),p}(z)=r(z^p)g_{f(x),p}(z^p)+b(z),$$

for some polynomials $r(z)$ and $b(z)$ depending on $f(x)$ and $p$. Such equations would place these generating functions in a Mahler-type functional framework; the paper gives specific examples and indicates that the general assertion remains conjectural.