Uniform boundedness conjecture for preperiodic points

Remarks

Here a point $P$ of projective space is preperiodic for $\phi$ when its forward orbit $\set{\phi^n(P) : n \ge 0}$ is finite, as in arithmetic dynamics. For each single $\phi$, the set of preperiodic points in $\mathbb{P}^N(K)$ is finite by Northcott's theorem on heights. The conjecture asks for a bound that depends only on $N$, $d$ and $D$.

The conjecture is open, even for the maps $\phi(z) = z^2 + c$ with $c \in \Q$ acting on $\mathbb{P}^1(\Q)$.

The case $N = 1$, $d = 4$ implies the uniform boundedness of torsion of elliptic curves, proved by Merel in 1996. Indeed, let $E$ be an elliptic curve over $K$. The $x$-coordinates of its torsion points are preperiodic points of the degree $4$ Lattès map that describes doubling on $E$. At most two points of $E$ share an $x$-coordinate.