Periodic Morton–Silverman conjecture over the rational numbers

Let $\varphi:\mathbb{P}^1(\mathbb{Q})\to\mathbb{P}^1(\mathbb{Q})$ be a morphism of degree $d$, and let $\operatorname{Per}(\varphi,\mathbb{P}^1(\mathbb{Q}))$ denote its set of periodic points. Fix an integer $d\geq 2$. Periodic Morton–Silverman conjecture. There exists a constant $C'=C'(d)$ such that, for every such morphism,

$$\#\operatorname{Per}(\varphi,\mathbb{P}^1(\mathbb{Q}))\leq C'(d).$$

This is the $(D,N)=(1,1)$ periodic specialization of uniform boundedness; unlike the corresponding finiteness statement for an individual morphism, uniformity over all degree-$d$ morphisms remains open.