BNCs provide a semblance of persistence. But the real problem with IRC is smartphone persistence, I think irccloud is the only service that solves this. But alas, it isn't free.
So can this be solved for free? Is it even a good idea to solve it?
thelounge is an web based IRC client with persistence like irccloud, except that thelounge is open source and self-hosted.
A web-based client is not going to provide adequate persistence on a smartphone.
Lounge works fine on a smartphone because the actual IRC connection would be handled by a server where you host it, not your own phone.
Do you get notifications when you are mentioned on your phone though?
Nobody has made persistent IRC channels that are as usable as newer chat products that don't try to be IRC.