Transcendence conjecture for Artin–Mazur zeta functions of separable rational maps

Let $p$ be a prime, let $f\in \overline{\mathbb{F}}_p(x)$ be a separable rational map, and let $\zeta(f,\mathbb{P}^1(\overline{\mathbb{F}}_p);t)$ denote its Artin–Mazur zeta function. Transcendence conjecture. The function

$$\zeta(f,\mathbb{P}^1(\overline{\mathbb{F}}_p);t)$$

is transcendental over $\mathbb{Q}(t)$. The paper proves rationality results for dynamically affine maps and observes that, in positive characteristic, rationality appears to occur almost exclusively for inseparable maps; the conjecture predicts transcendence in the separable case.