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Ok, with your encouragement I’ll give it a go.
Peter Scholze has replied to my MO question. That’s right my response that a stable object in a $\infty$-topos that is relative cohesive over condensed $\infty$-groupoids (or pyknotic ones anyway) should generate a fracture square?
I have removed the bulk of the entry, since that was a set of notes which has meanwhile evolved into polished form in arXiv:2008.01101, while what remained here had meanwhile become notationally inconsistent with both that and related $n$Lab entries, to the extent that it now would have been serious work to adjust notation.
So I just removed it for the time being, but added more prominent cross-links with the new entries singular cohesion and singular-smooth infinity-groupoid.