Condensedness of the gros Galois infinity-category

Determine whether the gros Galois infinity-category GAL, defined as the partially lax colimit of the condensed Galois infinity-categories Gal(X) over the category of kappa-small quasi-compact quasi-separated schemes with étale morphisms marked, is a condensed infinity-category, equivalently a sheaf on the category of kappa-small extremally disconnected compact Hausdorff spaces.

Background

The paper defines the gros Galois infinity-category GAL as the pointwise partially lax colimit of the condensed Galois infinity-categories Gal(X) over kappa-small quasi-compact quasi-separated schemes, with étale morphisms marked. The principal result of the section identifies Postnikov-complete étale sheaves on the category of such schemes with continuous functors from GAL to a condensed infinity-category associated to a compactly assembled presentable infinity-category.

The authors explicitly note that the proof of the gros étale exodromy theorem does not require GAL itself to be condensed. Whether GAL satisfies the sheaf condition, and therefore defines a condensed infinity-category, is left unresolved.