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Still think it'd be useful to exted the #ActivityPub client-server spec with some sort of standard for "searching" through your messages.
IMO this will be crucial for making clients that can be interoperable between implementations (which I think is important for user agency!)
@mauve I started this FEP a long time ago, but never had the energy to finish and submit it. See if it helps you with some ideas: https://git.sr.ht/~mariusor/Fediverse-Enhancement-Proposal/tree/master/item/feps/fep-6606.md
[edit]: I just pushed some reductions in scope, to make it more self contained.