DeMarco–Mavraki relative Dynamical Manin–Mumford conjecture

Let $S$ be a smooth and irreducible quasi-projective variety over $\mathbb C$, let $N$ be a positive integer, and let $\Phi:S\times\mathbb P^N\to S\times\mathbb P^N$ be an algebraic family of morphisms of degree greater than $1$. Let $\mathcal X\subseteq S\times\mathbb P^N$ be a complex irreducible subvariety flat over $S$, and let $r_{\Phi,\mathcal X}$ denote its relative special dimension. DeMarco–Mavraki's relative Dynamical Manin–Mumford conjecture. The following are equivalent: $\mathcal X$ contains a Zariski-dense set of $\Phi$-preperiodic points, and $\hat{T}^{r_{\Phi,\mathcal X}}_\Phi\wedge[\mathcal X]\neq0$. This is the relative version proposed for parametrized families; the paper proves only special cases, while the general conjecture remains open.