Read it again. 

The validation algorithm would be extended so that the transaction can't be replayed, because replaying it in the other network REQUIRES a child transaction in the same block that is valid, a child transaction the is unique to the network. By doing this policy change simultaneously in both networks, old pre-signed transactions *can not be replayed by anybody but the owner* of the coins (as he must spend them immediately in the child transaction). 

It means that as soon as spent, the UTXO sets immediately and irrevocably diverges across the two networks. Which is the entire point, isn't it?