Zariski-dense orbit conjecture

Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be an irreducible quasi-projective variety over $k$, and let $f$ be a dominant rational self-map of $X$. The condition that $f$ has no nonconstant invariant rational functions is

$$\{g\in k(X):g\circ f=g\}=k,$$

where $k(X)$ is the function field of $X$.

Zariski-dense orbit conjecture. There exists $x\in X(k)$ whose forward orbit under $f$ is well-defined and Zariski-dense in $X$.

The source presents a proof of this conjecture for a general split polynomial endomorphism of $(\mathbb P^1)^2$ whose factors all have the same degree $d\geq2$. The general statement supplied here is not resolved in the source, so its database status is open.