Dynamical uniform boundedness conjecture for function fields

Let $k$ be a field, let $K$ be the function field of an integral curve over $k$, and let $\operatorname{Rat}_d(\overline K)$ denote the degree-$d$ rational functions over an algebraic closure of $K$. The group $\operatorname{PGL}_2(\overline K)$ acts on rational functions by conjugation; write $\operatorname{PGL}_2(\overline K)\cdot\operatorname{Rat}_d(\overline k)$ for the resulting family of maps conjugate to degree-$d$ maps defined over $\overline k$. A family satisfies the Strong Uniform Boundedness Principle over $K$ if, for every $D\ge 1$, there is a bound depending only on the family and $D$ for the number of preperiodic points over every degree-$D$ extension of $K$. Dynamical Uniform Boundedness Conjecture for function fields. For each integer $d\ge 2$, the family

$$\operatorname{Rat}_d(\overline K)\setminus \operatorname{PGL}_2(\overline K)\cdot\operatorname{Rat}_d(\overline k)$$

satisfies the Strong Uniform Boundedness Principle over $K$. This removes the evident constant-field and conjugacy sources of counterexamples and is presented as the expected function-field analogue of the number-field conjecture; it remains open in the stated generality.