Relative sAND Conjecture for fiberwise endomorphisms

Let $X$ and $S$ be quasi-projective varieties over a number field $K$, and let $\pi:X\to S$ be a projective morphism. Let $f:X\to X$ be a surjective morphism satisfying $\pi\circ f=\pi$. Define $Z_f(d)$ as the bounded-degree locus of points whose relative arithmetic degree is smaller than the corresponding first dynamical degree. Relative sAND Conjecture. For every constant $d>0$, the set $Z_f(d)$ is not Zariski dense in $X_{\overline K}$. This is the relative, fiberwise analogue of the sAND conjecture; the paper proves equivalences and special cases but does not establish it in full generality.