Dynamical André–Oort conjecture for curves

Let $(f_t)_{t ein\Lambda}$ be a non-isotrivial algebraic family of rational maps with degree $d\geq 2$, parametrized by an algebraic curve $\Lambda$ over $\mathbb C$. A rational map is postcritically finite (PCF) if all its critical orbits are finite. Dynamical André–Oort conjecture for curves. The following are equivalent: there are infinitely many $t\in\Lambda$ such that $f_t$ is PCF; the family has at most one independent critical orbit. The conjecture concerns the distribution of PCF maps as special points in moduli spaces and was proposed as the curve case of the Dynamical André–Oort conjecture. The source paper states that it proves this conjecture, in fact establishing a stronger Bogomolov-type generalization for curves.