The Hilbertian-projective implies pro-free conjecture

Let $K\leq \overline{\mathbb Q}$ be a field. Call $K$ Hilbertian if it has the Hilbertian property, and projective if its absolute Galois group $G_K$ is projective, meaning that every surjective homomorphism to $G_K$ splits. A profree group is a profinite free group.

Hilbertian-projective implies pro-free conjecture. If $K$ is Hilbertian and projective, then $G_K$ is profree.

This conjecture concerns the structure of absolute Galois groups of algebraic extensions of $\mathbb Q$. The source attributes it to the cited work, but the material provided does not establish whether it has been resolved.