I tried searching for this, but I didn't find anything, so I apologize if I missed it.

Anyway, I know that the team with higher initiative will put up their passive barrier last, to override any other passive barrier that was put up.

However, what is the case for mystery barriers? Say, for example, there are two opposing Gaaras (NOT Kazekage) both in the first position of their teams. They both try to put up their barriers immediately on the 1st round. Which one goes second, and overrides the first? The one with higher or lower initiative?