The periodic-point version of Morton–Silverman uniform boundedness

Let $n\geq 1$ and $d\geq 2$ be integers. Let $K/\mathbb{Q}$ be a number field of degree at most $n$, and let $\varphi:\mathbb{P}^1(K)\to\mathbb{P}^1(K)$ be a morphism of degree $d$ defined over $K$. Write $\operatorname{Per}(\varphi,\mathbb{P}^1(K))$ for the set of periodic points of $\varphi$. Periodic uniform boundedness conjecture. There exists a constant $C'=C'(n,d)$ such that

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

This follows formally from the corresponding preperiodic-point bound if the latter is known. The paper studies this periodic formulation for polynomial maps over number fields and several local or function-field settings.