Push Notifications over LAN

This one is for the creative minds! I’ve been searching a lot but I cannot find what I want… I would like to send push notifications from a Home Assistant condition ( I can write an integration ) to my Flutter app, to run on Android an iPhones. There doesn’t seem a “normal” way to do push notifications without the internet, or without leaving a background process running on the phone. I wonder if someone found a way for this, or came across it and would like to share some thoughts.

My highest hopes were using Firebase CLI. It’s a sandbox that can run on a local server and provide some of the cloud services locally - you only have to login once. Unfortunately though it doesn’t have messaging, the service that allows for push notification. The need for push notification is required, to notify a device even when the app is not running.

Some wizard have some thought about this?