Arithmetic dynamical Mordell–Lang conjecture for $\mathbb{P}^1$

Let $X=\mathbb{P}^1$, and let $Y$ be a curve defined over a field $K$ of characteristic zero. Suppose that $\lambda:Y\to X$ is a finite $K$-morphism and that $\phi:X\to X$ is a morphism of degree at least two. For $a\in X(K)$, Arithmetic dynamical Mordell–Lang conjecture for $\mathbb{P}^1$. The set

$$\{n\geq 0:\phi^n(a)\in\lambda(Y(K))\}$$

is a finite union of arithmetic progressions. This is the arithmetic analogue proposed in the paper; the paper also discusses possible higher-dimensional generalizations, but gives no resolution of this statement.