Morton–Silverman Uniform Boundedness Conjecture for rational maps

Let $D,N\ge 1$ and $d\ge 2$ be integers. For a number field $K$ with $[K:\mathbb{Q}]=D$ and a morphism $\phi:\mathbb{P}^N\to\mathbb{P}^N$ of degree $d$ defined over $K$, a point of $\mathbb{P}^N(K)$ is preperiodic if its forward orbit under $\phi$ is finite. Uniform Boundedness Conjecture. There is a constant $B=B(D,N,d)$ such that every such morphism $\phi$ has at most $B$ preperiodic points in $\mathbb{P}^N(K)$. This is a major unsolved problem in arithmetic dynamics and generalizes uniform boundedness results for torsion points on elliptic curves.