Dynamical generalization of Siegel's theorem for integral points

Let $K$ be a number field with ring of integers $\mathfrak{o}_K$, let $\varphi$ be the dynamical system under consideration, and let $\mathbb{P}^1_{\mathfrak{o}_K}$ be the projective line over $\mathfrak{o}_K$. For points $\alpha,\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, say that $\alpha$ is integral relative to $\beta$ if the Zariski closure of $\alpha$ does not meet the Zariski closure of $\beta$ in $\mathbb{P}^1_{\mathfrak{o}_K}$. A point is preperiodic for $\varphi$ if its forward orbit under $\varphi$ is finite, and nonpreperiodic otherwise.

Dynamical Siegel conjecture. For any nonpreperiodic point $\beta\in\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$, there are at most finitely many preperiodic points of $\varphi$ in $\mathbb{P}^1_{\mathfrak{o}_K}(\overline{K})$ that are integral relative to $\beta$.

This is proposed as a dynamical generalization of Siegel's theorem for integral points. It is motivated by extending the paper's equidistribution results from periodic points and backward orbits to nonrepeating sequences of Galois orbits of preperiodic points; the supplied text does not give a resolution.