Periodic-point uniform boundedness conjecture over number fields

Let $\operatorname{Per}(\varphi,\mathbb{P}^1(K))$ denote the set of periodic points of a morphism $\varphi:\mathbb{P}^1\to\mathbb{P}^1$ defined over a number field $K$. Periodic-point uniform boundedness conjecture. Fix integers $D\geq1$ and $d\geq2$. There exists a constant $C'=C'(D,d)$ such that, for every number field $K/\mathbb{Q}$ of degree at most $D$ and every morphism of degree $d$ defined over $K$,

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

This is the $(D,1)$-version of the Morton–Silverman conjecture, obtained by counting periodic rather than all preperiodic points; it remains open.