Anti-transaction replay in a hardfork

The proposal suggests extending the validation algorithm to prevent replaying of transactions in both networks. This would be done by requiring a unique child transaction for each network, which would make it impossible for anyone but the owner of the coins to spend them immediately in the child transaction. As soon as the coins are spent, UTXO sets immediately and irrevocably diverge across both networks. In case Alice pays Bob with an old style time-locked tx, under the new rules, Bob can confirm the time-locked tx on both networks by sending the outputs to himself again with a different tx format. However, as Bob gets all the money on both forks, it is already a successful replay.