This feature is in default on email app for mobile. it is also called “Push Notifications .”
It have to introduce to desktop too. Thunderbird for desktop send notifications while it runs only.
Thunderbird
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Please make a lightweight system service to notify new e-mail and events, and open thunterbird on click
Currently i'm using Linux Ubuntu 24.04 with Gnome 46 and Windows 10 on another machine
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