Finite generation conjecture for Galois groups of PAC extensions

Let $K$ be a finitely generated field and let $M$ be an algebraic extension of $K$. A field is PAC over $K$ when it satisfies the PAC-over-$K$ property stated in the source.

Finite generation conjecture. If $M$ is PAC over $K$, then

$$\operatorname{Gal}(M)\text{ is finitely generated}.$$

This conjecture concerns the structure of absolute Galois groups of PAC algebraic extensions of finitely generated fields. The supplied text does not state whether it has been proved or disproved.