Height growth conjecture for subvarieties under iteration

Let $K$ be a number field and let $X$ be a smooth projective geometrically irreducible variety over $K$. Let $f\colon X\longrightarrow X$ be a surjective morphism, let $Y\subset X$ be a closed subscheme of codimension at least two, and let $H$ be an ample divisor. Fix global height functions $h_Y$ and $h_H$. For $x\in X(\overline K)$, let $e(Y)$ denote the ramification invariant associated with $Y$, and let $\alpha_f(x)$ denote the arithmetic degree. Height growth conjecture for subvarieties. If $e(Y)<\alpha_f(x)$ and the $f$-orbit of $x$ is Zariski dense, then

$$\lim_{n\to\infty}\frac{h_Y(f^n(x))}{h_H(f^n(x))}=0.$$

This is the global-height specialization of the preceding local-and-global conjectural statement. It is supported by the paper’s conditional theorem assuming Vojta’s conjecture; the associated orbit-intersection consequence is related to the Dynamical Mordell–Lang conjecture.