Uniform boundedness conjecture for rational maps of the form $z^d+g(z)$

Let $g(z)$ be any rational map, and for an integer $d$ define

$$f_{d,g}(z)=z^d+g(z).$$

For a number field $K$, let $\operatorname{Preper}(f_{d,g},K)$ denote the set of $K$-rational preperiodic points of $f_{d,g}$. Uniform boundedness conjecture. For every rational map $g$ and every positive integer $D$, there exists a constant $C(g,D)$ such that for every number field $K$ with $[K:\mathbb{Q}]=D$,

$$\#\operatorname{Preper}(f_{d,g},K)<C(g,D).$$

The conjecture reflects computational evidence for uniform boundedness in several families as the degree $d$ varies. The source gives no proof or counterexample.