Finiteness of integral points under pro-$p$ Galois hyperbolicity

Let $k$ be a finitely generated field and let $X$ be a hyperbolic algebraic curve defined over $k$. Let $S$ be a finite set of primes, assume that $Gal(\bar{k}/k)(p)$ is hyperbolic for every prime $p$ outside $S$, and let $A_S$ be the integral closure of $\mathbb Z$ in the maximal extension of $k$ unramified outside $S$. Pro-$p$ hyperbolicity finiteness conjecture. The number of points of $X$ defined over $A_S$ is finite. The source relates this to the section-conjecture philosophy and says it would yield Diophantine finiteness; no resolution is supplied.