Relative degree-stabilization conjecture for families of endomorphisms

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 given by $\Phi(s,p)=(s,F_s(p))$, where each $F_s$ has degree greater than $1$. Let $\mathcal X\subseteq S\times\mathbb P^K$ be irreducible, dominant and flat over $S$, with $\mathcal X_s$ preperiodic under $F_s$ for a Zariski-dense set of $s\in S$. Relative Dynamical Manin–Mumford conjecture. For every positive integer $N$,

$$\hat{T}^{\,r_{\Phi^{\times N},\mathcal X^N}}_{\Phi^{\times N}}\wedge[\mathcal X^N]\neq0.$$

This is a conjectural extension of the paper's special case from regular polynomial endomorphisms of $\mathbb P^2$ to arbitrary projective dimension and families; it remains open in most cases.