Hello @anonymee, thank you for the contribution.
Your suggestion is more like a notification system to know when a user receives a message and aiding prompt reply.
However, you should note that Lichess is an internet chess application and not a social media application.
The app developers have done well to include the Inbox feature and also provide the unread messages when the user menu is opened.
According to this PR here, it shows that the PR does not include a notification system for new messages which your suggestion falls into.
Therefore, it clearly shows that the app developers/PO know about the absence of this feature but decide to leave it like that since you can get the unread messages by just navigating to the inbox section of the application.
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