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In order to obtain and view the Supreme Crest, complete every single side mission before starting the Final Chapter (you should be on 97%).
After completing the first two levels in the Final Chapter, you should have 100% complete.
Create a new save file and then die during the boss fight. It will ask if you want to try again. Say no.
Go back to the airship and voila, you can apply the Supreme Crest to the airship! And then you have one save file for free exploring and one right before the boss battle!