The finite-ramification characterization of postcritically finite morphisms

Let $X$ be a smooth, irreducible, projective variety defined over the field of fractions $K$ of a Dedekind domain $R$. Let $\varphi:X\to X$ be a morphism defined over $K$, let $\alpha\in X(K)$, and let $S_\infty(\alpha)$ denote the set of primes of $K$ ramified in the field generated by all iterated preimages of $\alpha$. Assume that $\varphi$ is non-PCF and that

$$\bigcup_{n>0}\varphi^{-n}(\alpha)$$

is infinite. The finite-ramification characterization conjecture. Then $S_\infty(\alpha)$ is infinite. This conjecture reverses the finite-ramification theorem for PCF morphisms; it is known in the stated source only for number fields with $X=\mathbb{P}^1$, while the general higher-dimensional case remains open.