The wild-ramification conjecture for preimage fields

Let $X$ be a smooth, irreducible, projective variety over the field of fractions $K$ of a Dedekind domain $R$, let $\varphi:X\to X$ be a PCF morphism defined over $K$, and let $\alpha\in X(K)$ lie in the setting of the finite-ramification theorem. Write $S_\infty(\alpha)$ for the finite set of primes of $K$ ramified in the field generated by all iterated preimages of $\alpha$. Wild-ramification conjecture. There is at least one prime of $K$ in $S_\infty(\alpha)$ at which the ramification is wild. This conjecture proposes that the finite ramification supplied by the PCF theorem can never be entirely tame; its resolution is not given in the source.