The Medvedev–Scanlon conjecture on dense forward orbits

Let $X$ be an irreducible variety defined over an algebraically closed field $F$ of characteristic $0$, and let $\phi\colon X\dashrightarrow X$ be a dominant rational map. A dominant rational map is fiber preserving if there is a positive-dimensional variety $Z$ and a dominant rational map $\psi\colon X\dashrightarrow Z$ such that $\psi\circ\phi=\psi$. Medvedev–Scanlon conjecture. If $\phi$ is not fiber preserving, then there is a point $x\in X(F)$ with a forward dense orbit under $\phi$. This conjecture predicts a dynamical criterion for the existence of a point with dense orbit, and connects orbit structure with rational fibrations preserved by the map. Its resolution status is not specified in the supplied source context.