If the arrows are for scrolling the tabs horizontally, what's the purpose of having tabs?
You won't notice when someone sends you a message if it's past the scroll line because the tab will not flash, and you won't be able to see all your conversations automatically.
I think a better way to handle this would be to auto-strech the tabs to fit as many as possible on one line until only the first four or five letters of the email/nickname/custom name can be seen, and then, if more space is needed, use two rows. This is how Avant Browser and many other tabbed UIs handle this, and in my opinion it works much better than having to click 5 times a button labeled ">" to see who I was talking with.
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