I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)=\mathrm{Aut}\underline\pi^{temp}_{{\mathbb{C}}_p}$), conditional on Pop's theorem. In there, it is referenced as unpublished at the time (2002):
- F. Pop, A combinatorial description of $G_\mathbb{Q}$, talk at Oberwolfach, July 1998 (cf. Tagungsbericht 28), manuscript in preparation.
A report of the 1998 Oberwolfach conference survives. The relevant section reads:
Let $\mathcal{Var}_\mathbb{Q}$ be the category of all $\mathbb{Q}$-varieties and morphisms of such varieties. Taking the fundamental group functor we get
$$\overline{\pi}_1: \mathcal{Var}_\mathbb{Q} \to \mathcal{G}$$ $$X \mapsto \pi_1(\bar{X})$$
into the category $\mathcal{G}$ of all profinite groups and outer homomorphisms. Let $\mathcal{G}_\mathbb{Q}$ be the image of $\mathcal{Var}_\mathbb{Q}$ under $\overline{\pi}_1$. For every $X$ there exists a canonical representation $$\rho_X:G_\mathbb{Q} \to \mathrm{Out}(\pi_1(\bar{X}))=\mathrm{Aut}_\mathcal{G}(\pi_1(\bar{X}))$$ which behaves functorially. Then we get a homomorphism
$$\imath_\mathbb{Q} : G_\mathbb{Q} \to \mathrm{Aut}(G_\mathbb{Q})$$ $$\sigma \mapsto (\rho_X(\sigma))_X$$
We gave a positive answer to the question of Oda-Matsumoto, asking whether $\imath_\mathbb{Q}$ is an isomorphism.
It seems like Pop didn't publish anything else about this result.
Is the proof (or a sketch of it) available anywhere?