Periodic-point density conjecture for finite-field quadratic polynomials

Let $f\in\mathbb{F}_q[x]$ be a monic quadratic polynomial, and say that two polynomials are conjugate when they are related by the relevant change of coordinates. Let $\operatorname{Per}(f)$ be the set of periodic points of $f$, and let $\delta$ denote the density defined in the surrounding discussion. Periodic-point density conjecture. If $f$ is not conjugate to $x^2$ or $x^2-2$, then

$$\delta(\operatorname{Per}(f))=0.$$

The conjecture is motivated by arboreal representations and techniques that do not follow directly from the maximality theorem in the quadratic case; its status is unresolved in the source.