Ghioca–Tucker's dynamical Mordell–Lang conjecture for rational self-maps

Let ${\mathbf{k}}$ be an algebraically closed field of characteristic $0$, let $X$ be a variety defined over ${\mathbf{k}}$, and let $f:X\dashrightarrow X$ be a rational self-map. Say that $(X,f)$ has the DML-property if, for every subvariety $V$ of $X$ and every $x\in X({\mathbf{k}})$ whose orbit is well-defined, the set

$$\{n\geq 0\mid f^n(x)\in V\}$$

is a finite union of arithmetic progressions. Dynamical Mordell–Lang conjecture. The pair $(X,f)$ satisfies the DML-property. This generalizes the dynamical Mordell–Lang conjecture to rational self-maps; it is known in a number of special settings but remains open in the stated generality.