Hi KamilZ,
a workaround would be to nevertheless disable the (automatically appearance of the) “Start Window”
(Tools > Customize… > [Window] > [Customize Start Window…] >> [_] Always show…… )
and use the Arrow-Down key instead to open it on your own on (empty / non-dirty) Untitled documents.
Perhaps that would be something for you and solve your problem with the Tray Icon?