There is a lot of confusion here in this topic. Let’s start over:
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Discourse Hub App
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Using discourse.org hosting: You can receive push notification when you add the hosted site to the Discourse Hub app, be it Android or iOS.
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Not using discourse.org hosting: You won’t get push notification from the Discourse Hub app.
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Android Google Chrome
- You can enable push notification from the browser. You will be prompted to do so in a banner after posting your first reply and having TL1. You can also enable it anytime in your profile. You don’t need to add the app to the homescreen to do so (for now).