Positive-characteristic dense-orbit conjecture over transcendental fields

Let $k$ be an algebraically closed field of characteristic $p>0$ that is transcendental over $\mathbb{F}_p$. Let $X$ be a quasi-projective variety over $k$ and let $f:X\to X$ be a dominant endomorphism. Assume that there is no nonconstant rational function $g$ satisfying $g\circ f=g$.

Positive-characteristic dense-orbit conjecture. There exists a point $x\in X(k)$ whose orbit is Zariski dense in $X$.

The source explains that the analogous assertion fails over the algebraic closure of a finite field because all point-orbits are finite, and proposes this transcendental-field version as the expected replacement.