Hindry's integral preperiodic-point conjecture for rational maps

Let $k$ be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $R(x)\in k(x)$ be a rational function of degree at least $2$, and consider the dynamical system associated to the rational map $R_*:\mathbb P^1\to\mathbb P^1$. Let $\alpha\in\mathbb P^1(\overline{k})$ be non-preperiodic for $R_*$. A pre-periodic point is $S$-integral with respect to $\alpha$ when its Zariski closure in $\mathbb P^1/\operatorname{Spec}({\mathcal O}_{k,S})$ does not meet the Zariski closure of $\alpha$. Hindry's conjecture. There are only finitely many pre-periodic points $\xi\in\mathbb P^1(\overline{k})$ that are $S$-integral with respect to $\alpha$. The conjecture is equivalent to the elliptic-curve theorem in the case of Lattès maps, and the paper proves it for $R(x)=x^2$; the general assertion, including the Chebyshev and broader dynamical cases, remains open in the supplied text.