Essential Image of the Étale Homotopy type

For any scheme $$X$$ we can associate the étale homotopy type $$Et(X)$$, which is a pro-object in the homotopy category of CW-complexes. My question is, do we have a good understanding of the essential image, i.e., do we have a set of necessary and sufficient conditions on a pro-CW-complex for it to be the étale homotopy type of some scheme. Even more "ambitiously", do we have such a set of conditions such that a pro-CW-complex is the étale homotopy type of a smooth projective scheme $$K$$-scheme?

• I am not sure if one of these tasks is more ambitious than the other. It seems unlikely that you could characterize all the possible fundamental groups for smooth projective $\mathbb{C}$-schemes. By the way what is $K$ in your question? Depending on whether it is separably closed or not we are dealing with rather different problems). – user145520 Aug 3 '20 at 8:31
• What is your construction of the homotopy type for non-locally Noetherian schemes? – user145520 Aug 3 '20 at 8:32
• @vrz The étale homotopy type is perfectly well defined for any scheme, by taking the shape of the étale topos, although probably considering only the profinite shape would make the question more approachable. – Denis Nardin Aug 3 '20 at 9:01
• @vrz I'm happy to assume that $K$ is spearably closed and characteristic zero, if that helps in any way. – curious math guy Aug 3 '20 at 11:16