Baker's conjecture on S-integral preperiodic points

Let $K$ be a number field, and let $S$ be a finite set of places of $K$ containing all archimedean places. Let $\varphi:\mathbb{P}_K^{1}\to\mathbb{P}_K^{1}$ be a nonconstant rational function of degree $d\geq 2$, and let $\alpha\in\mathbb{P}^{1}(K)$ be non-preperiodic for $\varphi$. A point $\beta\in\mathbb{P}^{1}(\bar{K})$ is $S$-integral with respect to $\alpha$ if no conjugate of $\beta$ meets any conjugate of $\alpha$ at primes lying outside $S$. Baker's conjecture. There are at most finitely many preperiodic points $\beta\in\mathbb{P}^{1}(\bar{K})$ that are $S$-integral with respect to $\alpha$. The conjecture predicts finiteness of integral relations between preperiodic points and a non-preperiodic point for rational maps; the paper proves uniformity results for Chebyshev polynomials, while the general statement remains open.