@julian @JessTheUnstill After a brief dive into the mastodon API documentation, it seems to be a protocol limitation indeed. You (or rather your client program) do not get to see the conversation ID until you join that conversation, so until then, you do not have the correct piece of information to request muting of that conversation.
Some references:
https://docs.joinmastodon.org/methods/timelines/#home
https://docs.joinmastodon.org/methods/conversations/#get



