In Illusory 106, 5F, you'll find a mirage in a dead-end with two chests. Talk to her to start this quest. For this quest, you have to hunt the various higher-level mirages in this floor and obtain their performa. Take note that since there are new enemies in this floor beyond, you'll probably obtain performas that can be used to unlock new Carnage and Radiant unities for your characters.
You can keep track of your collected Performa by opening the Items menu and switching to the Performa tab. Here are the list of mirages that you should defeat and collect the required performas from.
Upon collecting the necessary performa, return to the quest giver and surrender the items to complete the quest. You'll receive another Treasure Key which you can use to unlock the nearby chest.