What's the URL of the page this enhancement relates to, if any?
Any review request page.

Describe the enhancement and the motivation for it.
The Push Notification API allows users the ability to give websites permissions to send them updates about things they care about, even when they've closed the tab to that page.
Review Board could let the user opt-in to push notifications for getting updates about their review requests (or any review request they've been conversing in). The notification, when clicked, would open a browser tab to that review request page at the most recent update.
See: https://support.mozilla.org/en-US/kb/push-notifications-firefox

and: https://developer.mozilla.org/en-US/docs/Web/API/Push_API
Also see: http://serviceworke.rs/
This might be worth first implementing as an Extension to experiment with it.