But I've seen nothing that suggest the quest will be repeated, just the FATE's.
They specifically stated the event would be repeated. The ps4 version will be in beta by then and the ps4 players deserve a chance to get the gear.

If the quest would be re-triggered then surely people would end up with two sets of gear (the same problem with the AF quests).
The AF quests and gear don't differ by gender. The male and female Lightning gear are all seperate items with different names. It checks your character's gender when you take the quest, otherwise it wouldn't know what to show as the reward.

I understand you being cynical, but i'm not talking out my behind here.