You have concluded your business in Il Mheg, and Thancred suggests checking in on the Crystarium.
This quest is available if you have completed the quest Acht-la Ormh Inn.
Starting the Quest
Talk to Thancred in Il Mheg to begin the quest.
Speak with Alisaie in the Crystarium
Teleport to the Crystarium, and use the Aetheryte to go to the Dossal Gate. Go to the quest icon on the map and talk to Alisaie to proceed. There will be a cutscene. This one is important, so it is recommended to watch it either now or via the Unending Journey in an inn room.
Speak with Alisaie
Talk to Alisaie to proceed.
Speak with the Manager of Suites
Use the nearby Aethernet Shard to go to the Pendants. Go up the stairs and through the archway to the quest icon on the map. Talk to the Manager of Suites and say Yes to complete the quest. There will be a cutscene.
Next Main Scenario Quest
After you complete the quest, you will need to be at least level 74 to proceed to the next quest. After you reach level 74, talk to the Manager of Suites to accept the next quest: A Party Soon Divided.