Tbh the way you show and explain it makes it feel annoying to the sight when you're playing besides that it would need of a key to switch between tabs and no one would focus on that outside training rooms, in any case it can be programmed that received and sent whispers appear in the general chat list in lobby when you leave a room or in the mailbox window with a mark that shows that you received a whisper.
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Although in options > misc you can block the chat that you wanna ignore, whisper, team, general, all 3... I dislike this idea.
