The dynamical-degree-one alternative for invariant subvarieties

Let $X$ be a smooth projective variety over an algebraically closed field $K$ of characteristic $0$, and let $\phi:X\dashrightarrow X$ be a dominant rational self-map. Assume that no non-constant rational function $f:X\dashrightarrow\mathbb{P}^1$ satisfies $f\circ\phi=f$. The dynamical-degree-one alternative. Exactly one of the following holds: the dynamical degree of $\phi$ equals $1$, or the union of all $\phi$-invariant proper subvarieties of $X$ is Zariski dense. This is stated as an even stronger related conjecture; the source notes that any counterexample to it, or to the preceding strengthening, must have dimension at least three.