Hyperbolicity conjecture for absolute Galois groups of number fields

Let $k$ be a number field, and let $Gal(\bar{k}/k)$ denote its full Galois group. Absolute Galois group hyperbolicity conjecture. For any number field $k$, the profinite group $Gal(\bar{k}/k)$ is hyperbolic. This is presented as the basic conjecture underlying the proposed applications to Diophantine finiteness; no resolution is supplied in the source.