Almost locally free fields

Using the positive solution of the general Abhyankar's conjecture, we prove that the fundamental group π₁(C) of the smooth connected affine curve C is "almost free". That is, for each positive integer e and for almost all σ=(σ₁,...,σ_e)επ₁(C)^e in the sense of the Haar measure, the closed subgroup of π₁(C) generated by σ₁,...,σ_e is profinite free on e generators. This implies a theorem of Harbater-Stevenson, proved by other means, that every finite embedding problem for π₁(C) is solvable, if we restrict the problem to a suitable open subgroups.