After completely finishing Storms of Fate (including the final, post-battle cutscene in Ru'Lude Gardens), you must wait until the next game day to activate this quest. Please note that your in-game Mission tracker will not update your mission progress until The Last Verse is completed. It will show that you're still on Dawn and Awakening even if you've completed Storms of Fate, so you will need to keep track of your own mission progress.
This is a separate cutscene from the one that concludes Storms of Fate.
Visit the three Promyvions and get the following key items at the Memory Fluxes on the final islands (4th floor). You can ascend each floor by looking for the pavillion-like structures with a giant glowing ball inside them. When you kill the glowing balls, there is a chance that it will spawn a teleport that takes you up to the next floor. There are Memory Fluxes on the 3rd floor as well but they are NOT the correct one. The higher floors have monsters that aggro but are easily dispatched at level 99. You can also call Trusts in this zone.
To get to Promyvion - Holla, go to La Theine Plateau in Zulkeim using the Survival Guide or if you are a White Mage use Teleport Holla, the entry is the Shattered Telepoint (J-8), once you enter into the Hall of Transference go to the Cerment Door at the far end of the chamber and interact with it and you will be pulled into Promyvion Holla.
To get to Promyvion - Dem, go to Konschat Highlands in Zulkeim using the Survival Guide or if you are a White Mage use Teleport Dem, as with Crag of Holla, go to the Shattered Telepoint (I-7) and interact with it to enter the Crag of Mea's Hall of Transference and interact with the Cerment Door at the far end and get pulled into Promyvion Dem.
To get to Promyvion - Mea, go to Tahrongi Canyon in Kolshushu using the Survival Guide or Teleport Mea if you are a White Mage and head for the Shattered Telepoint (I-6) and enter the Hall of Transference using the Shattered Telepoint and then like the other two go to the Cerment Door within this chamber and you will be pulled into Promyvion Mea.