In Ubuntu 16.04 I had a Thunderbird icon in tray bar with unread messages but there isn't in Ubuntu 18.04.
Is there a chance to have it again?
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Try Birdtray (https://github.com/gyunaev/birdtray)
Install on Ubuntu 18.04:
$ sudo add-apt-repository ppa:linuxuprising/apps $ sudo apt install birdtray