After you and Lyse successfully complete the trial in Bardam's Mettle, you wait for Hien and Gosetsu to return.
This quest is available if you have completed the quest In the Footsteps of Bardam the Brave.
Starting the Quest
Talk to Lyse in the Azim Steppe to begin the quest. 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 Magnai
Before you proceed, you may wish to attune to the Aetheryte outside. Afterward, talk to Magnai to complete the quest.
Next Main Scenario Quest
After you complete the quest, you will need to be at least level 66 to proceed to the next quest. After you reach level 66, talk to Baatu to accept the next quest: The Labors of Magnai. There will be a cutscene.