Finiteness conjecture for N-admissible rational functions

Let $N$ be a positive integer. A rational function $R(x)\in\mathbb{C}(x)$ is $N$-admissible if both $R(x)$ and $1-R(x)$ have the form

$$C x^d\prod_{i=0}^{N-1}(x-\mu^i)^{c_i},\qquad \mu=\exp(2\pi i/N),\quad C\in\mathbb{C},\quad d,c_i\in\mathbb{Z}.$$

Finiteness conjecture. For every positive integer $N$, the number of $N$-admissible rational functions is finite. The conjecture is stated as plausible without a rigorous proof in the source; the case $N=1$ is described as easy.