Higher Kawaguchi–Silverman conjecture for subvariety arithmetic degrees

Let $X$ be a smooth projective variety, let $f\colon X\dashrightarrow X$ be a dominant rational map, and let $V\subset X$ be a subvariety of dimension $k$. Let $\alpha_{k+1}(f;V)$ be the arithmetic degree associated with the orbit of $V$, and let $\alpha_{k+1}(f)$ be the corresponding invariant of $f$. Higher Kawaguchi–Silverman conjecture. One has

$$\alpha_{k+1}(f;V)\leqslant\alpha_{k+1}(f).$$

If $V$ has Zariski dense orbit, then

$$\alpha_{k+1}(f;V)=\alpha_{k+1}(f).$$

This predicts that the arithmetic height growth of a subvariety is bounded by the global higher arithmetic degree, with equality for a Zariski-dense orbit; the paper states it as an analogue of the pointwise Kawaguchi–Silverman conjecture.