The extroverts make the first move in the battle and the introverts don't respond to them, come what may. So the extroverts win.
The introverts, inside their mind have a battle of self talk with the extroverts. They want the extroverts to be a bit patient, less reactive, keep their feelings to themselves, just shut up.
Since the battle is being played in their own mind, they own the battlefield and guess who wins? The introverts win.
So yes there is a covert battle being played between the introverts and extroverts.