Would be amazing if it did!

There was a proposal for a NIP to allow relays - or side-servers - to offer push notifications. However, on iOS this would go through Apple App Store servers and on Android through Google's Cloud Messaging (GCM) - and if you were to only have local notifications, you'd constantly have to run the app in background, which isn't very battery efficient...

なんか許可聞かれますね

私は拒否してある