Seem to get 1 per week for doing the weekly islands round, so 3 weeks to find out what happens next. Don't hold your breath. ;)
Edit: Next step is a mission on your ship. I kept looking for a quest in log, or an item in my bags after doing my third week of islands, could not find any. Just a mission added to the mission board on the ship.
Quest
Worldvein Rumors
Quest facts and objectives from the current retail game data.
The highest voted Wowhead comments for this page. They are community notes, not TheWoWDB comments.
This is for your Worldvein Renesonce Rank 3. Once per week is a little stiff, but to be fair it is for a rank 3 essence.
EDIT: To add to this, you get a 3 day mission, followed up my another 6 day mission, followed by a 16 day mission. They can be countered and lowered to something better, but if you do it at the full time frame it will take 6+ weeks to obtain, or almost 4 weeks with reduction to your missions.
Maybe that is not quite fair after all.
If your quest giver has Zoned out it is because there is a second Flynn in Boralus that is tied to a quest and that NPC needs to be removed. I acheived this by starting The Ashvane Trading Company quest. After starting said quest my Quest giver appeared again and I was able to turn it in.
Quest facts
The client quest POI data records no map location for this quest, so no map is shown.
Check whether you have already completed this quest
in-game: /run print(C_QuestLog.IsQuestFlaggedCompleted(56533))
Description
Gather intelligence to assist SI:7 in identifying the location of the Fluctuating Worldvein.
Community notes from Wowhead · 44 comments
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Spell: Investigating the RumorsOpens up the mission Investigating the Rumors on your Mission Board.