Semi-freeness conjecture for absolute Galois groups of fully Hilbertian fields

Let $K$ be a fully Hilbertian field, and let $G_K$ denote its absolute Galois group. A profinite group of infinite rank $m$ is semi-free if every finite split embedding problem for it has a set of $m$ independent solutions. Semi-freeness conjecture. The absolute Galois group $G_K$ is semi-free of rank $|K|$. This conjecture predicts a strong group-theoretic consequence of full Hilbertianity, relating the arithmetic abundance of specializations over $K$ to independent solutions of finite split embedding problems. The source gives no resolution, so the conjecture remains open.