The Hilbertianity conjecture for projective absolute Galois groups

Let $K\subset\overline{\mathbb Q}$ and suppose that the absolute Galois group $G_K$ is a projective profinite group. The group $G_K$ is profree if it is a free profinite group. Hilbertianity conjecture. Then, $K$ is Hilbertian if and only if $G_K$ is profree. This generalizes the relevant form of Shafarevich's conjecture and is connected to the use of Hilbert irreducibility in regular inverse Galois theory; the source notes partial implications and a result under a pseudo-algebraically closed hypothesis.