Galois-conjugate projective classifying spaces conjecture

Let $X$ be a projective $K(\pi,1)$, meaning a projective variety whose universal cover is contractible, and let $\sigma\in\operatorname{Aut}(\mathbb{C})$. Write $X^\sigma$ for the variety obtained by applying $\sigma$ to the coefficients of equations defining $X$. Galois-conjugate classifying spaces conjecture. The conjugate variety $X^\sigma$ should still be a classifying space $K(\pi',1)$ for some group $\pi'$. This is motivated by the fact that the property holds for all known examples and that Galois conjugation preserves several related invariants, including products, isogeny, and algebraic fundamental groups; its general validity is not established.