Maintaining a persistent WebSocket connection would translate to poor battery life. And as a user, I would prefer that background web apps respect my battery life and not waste it.
I feel like the better approach would be to register for push notifications via your Service Worker, and send a push notification when you want to notify the user of something. When they follow the notification back to your web app, you can re-connect your WebSocket connection if it makes sense for your use case.
There is another use case: where you want to notify your app without notifying the user. However, I feel this can be pretty user-hostile, and browser vendors would not want to provide APIs that allow developers to be sneaky. As a browser user, I expect browser vendors to try to protect me from sneaky web apps.