DeMarco–Mavraki dynamical unlikely-intersection conjecture

Let $S$ be a smooth irreducible quasi-projective variety over $\mathbb{C}$, let $\Phi:S\times\mathbb{P}^N\to S\times\mathbb{P}^N$ be an algebraic family of endomorphisms of degree $d$, and let $\mathcal{X}\subseteq S\times\mathbb{P}^N$ be a closed irreducible subvariety flat over $S$. Define $\Phi$-special subvarieties using a polarizable endomorphism of a subvariety of the generic fiber, and let $r_{\Phi,\mathcal{X}}$ be the minimum relative dimension of a $\Phi$-special subvariety containing $\mathcal{X}$. Let $\widehat{T}_{\Phi}$ be the dynamical Green $(1,1)$-current associated to $\Phi$. DeMarco–Mavraki's conjecture. The following are equivalent:

1. $\mathcal{X}$ contains a Zariski dense set of $\Phi$-preperiodic points.

$$\widehat{T}_{\Phi}^{\wedge r_{\Phi,\mathcal{X}}}\wedge[\mathcal{X}]\neq 0.$$

This is presented as a dynamical generalization of the Relative Manin–Mumford conjecture. The supplied text does not establish the conjecture itself; it only proves a special case, so its general status remains open.