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@saal so me and Maximilien Peroux have recently proven the equivalence for modules over ΩX and comodules over (connected) X in any ∞-topos, but it's harder for general ∞-categories
@SamGunningham if you're interested in, I dunno, collaborating in some fashion on this type of thing, please send me an e-mail. Sounds like you have a lot more clarity about it than I do. :-)
