The kingdom of Italy had found itself in a remarkably poor position following Bruno's rapid advance through its northernmost regions. Hours was all it took for the German Army and its allies to seize their objectives.
It was entirely possible that the Italian soldiers defending those regions did not even fight, and instead rose the white flag the moment they made contact with the enemy. And by the time a treaty had been agreed upon, this was proven to be true in multiple cases.
Take Milan, for example, where Bruno and his soldiers had advanced from the Alps down to their beloved and ancient city. The Italian soldiers threw down their rifles and waved the white flag the moment they saw the thousands of armored vehicles rolling their way.