Zhang's relative dynamical Manin–Mumford conjecture for families

Let $S$ be a smooth irreducible quasi-projective variety over ${\mathbb C}$, let $K$ be a positive integer, and let

$$\Phi:S\times {\mathbb P}^K\to S\times {\mathbb P}^K$$

be a family of endomorphisms with $\Phi(s,p)=(s,F_s(p))$, where each $F_s$ is an endomorphism of ${\mathbb P}^K$ of degree greater than $1$. Let $\mathcal{X}\subseteq S\times {\mathbb P}^K$ be irreducible, dominant over $S$, flat over $S$, and suppose that $\mathcal{X}_s$ is preperiodic under $F_s$ for a Zariski dense set of $s\in S$. For a positive integer $N$, let $\Phi^{\times N}$ and $\mathcal{X}^N$ denote the $N$-fold fiber products, and let $r_{\Phi^{\times N},\mathcal{X}^N}$ be the corresponding relative special dimension. Zhang's conjecture. For every positive integer $N$,

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

The conjecture is presented as a consequence of the DeMarco–Mavraki conjecture and extends the constant-family result that a family containing a Zariski dense set of periodic curves must itself be periodic. The source gives no resolution status.