Though I don't see Patchou replying here anymore, I have some idea of tab conversation for Patchou to consider. I lazily made a picture in Paint for this.
The most left button is the open a new tab button. The most right button is the close tab button. The second right button is the moving current tab out of this window. And the last ones, the third right button are the up and down buttons to view the upper/lower group of tabs, similar to scroll keys in places like web browser. Anyway, the dark blue up arrow indicates that you can scroll up, while the light blue down arrow indicates that you can scroll down.
I actually think that this way is actually very convenient and does not waste much space. I hope Patchou will read this...