Baker–DeMarco's dynamical André-Oort conjecture

Let $(f_t)_{t\text{{?}}}$ be a non-isotrivial algebraic family of rational maps of degree $d\geq 2$, parametrized by an irreducible algebraic variety $\Lambda$ over $\mathbb{C}$ of dimension $N$. A family has independent critical orbits when the critical orbits satisfy the independence condition used in the source.

Dynamical André-Oort conjecture. The following are equivalent:

1. $f_t$ is PCF for a Zariski dense subset of $t\in \Lambda(\mathbb{C})$;
2. the family has at most $N$ independent critical orbits.

This conjecture describes the distribution of postcritically finite maps as dynamical special points in moduli spaces. The supplied passage identifies it as a conjecture of Baker and DeMarco and notes that the curve case has been solved, while the general statement remains unresolved.