There was a standard proposal for using centralized push server in Dovecot: coi-specs/webpush-spec.md at master · coi-dev/coi-specs · GitHub
The source code for Dovecot plugin is here: GitHub - dovecot/coi: Chat over IMAP plugin for dovecot
COI was a Delta Chat fork with such kinds of Dovecot integrations, now discontinued.
But I am not convinced this kind of centralized push server is needed at all. We all somehow get away without using this kind of “service” on our laptops, and Delta Chat for Android works well for me maintaining its own connection.
One of the valid reasons mobile OS vendors are making developers to use centralized push notifications is to prevent low quality apps from “app stores” from polling their servers every 5 seconds in the background and eating the battery. But as long as the app is using a proper “push” strategy, like Delta Chat with IMAP IDLE, there is no reason to switch to centralized server unless you have 50 messengers that actively push new messages to you in parallel, improvement in battery life is close to zero.