For those who want to replay the game’s puzzles and story without loosing all of their collectibles and health upgrades. This is a secret option that the game does not tell you exists.
Steps
1. Complete the main story. If you want NG+, you obviously have to have completed the final stage already.
2. Select Treasure Hunter Mode. While this should place you in an older level, all enemies and puzzles will be completed. This is not what we want, but it is a required step.
3. While in a Treasure Hunter Mode stage, pause the game and select New Game. You will be asked to select your difficulty as if you were starting over completely.
4. Starting a New Game in this way, you will be able to check your PDA and see that even though you’ve started the story all over again, you have kept all of your collectibles and Relics as well. Now you can overwrite your save if you see that you kept everything you collected before.