Originally Posted by Ultrapipi

This is a misconception.

If we follow DMM then...

That is... pretty much exactly that OP said.
BB winning gets us second 6* BB. 5* on second/third place gets us second 5*. But should lower rarity girl be in top three, her new version (5*/6*) will be higher rarity than original. Upgrade!
In the same vein, 6* in 2nd/3rd place will make her new version have worse rarity than original. Downgrade!