The higher arithmetic-degree Kawaguchi–Silverman conjecture for subvarieties

Let $X$ be a projective variety over a number field $K$, let $f:X\dashrightarrow X$ be a dominant rational self-map, and let $V$ be an irreducible subvariety of dimension $k$. Suppose the $f$-orbit of $V$ is well-defined. Write $\alpha_{k+1}(f,V)$ for the higher arithmetic degree of $V$ under $f$, and $\alpha_{k+1}(f)$ for the corresponding arithmetic degree of $f$. Kawaguchi–Silverman conjecture for subvarieties. The following assertions should hold: (1) the limit defining $\alpha_{k+1}(f,V)$ exists; (2) if $V$ has Zariski dense orbit, then

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

This was proposed as a higher-dimensional analogue of the Kawaguchi–Silverman conjecture for points. The second assertion is false, so the combined conjecture is refuted.