While pondering the impasse that you find yourselves in, you have an unexpected encounter outside of the Rostra.
This quest is available if you have completed the quest A Guide of Sorts.
Starting the Quest
Talk to Alisaie in Old Sharlayan to begin the quest. There will be a cutscene.
Speak with Alphinaud
Talk to Alphinaud, and he will accompany you.
Speak with Alisaie
While Alphinaud is accompanying you, talk to Alisaie and she will join you.
Speak with Ameliance
Go east down the stairs and go to where the orange arrow is pointing in the minimap. On the way, there is an optional glowing spot on the bridge where you can talk to Alphinaud and Alisaie about the view from the bridge. Continue going east, and if you have not attuned to the Aethernet Shard yet, you may wish to do so. Then go to where the orange arrow is pointing in the minimap and speak to Ameliance to proceed. There will be a cutscene.
Speak with Alphinaud Again
Go south to the quest icon in the minimap and talk to Alphinaud to proceed.
Speak with Krile
You can use the nearby Aethernet Shard to go to the Baldesion Annex if you have attuned to it. Go into the Baldesion Annex and check on the Main Hall Entrance, and say Yes to enter the main hall. Talk to Krile to complete the quest.