Morton–Silverman uniform boundedness conjecture for preperiodic points

Let $d\geq 2$, $N\geq 1$, and let $k$ be a number field. Define

$${\rm Preper}(f,k)=\left\{x\in {\mathbb P}^N(k):x\text{ is $f$-preperiodic}\right\}.$$

Morton–Silverman conjecture. There exists a constant $C=C(d,N,[k:{\mathbb Q}])>0$ such that every degree $d$ endomorphism $f$ of ${\mathbb P}^N$ over $k$ satisfies

$$\# {\rm Preper}(f,k)\leq C.$$

This is the number-field analogue of the geometric uniform boundedness conjecture. The source discusses substantial partial results and explicit bounds in special families, while the general statement remains open.