I have completed the quest on multiple characters. The trick was if in a group only the leader can do the quest. You talk to the quest giver and get the quest. Before killing anything you must "chat" with her and say I am ready (or something similar) and that starts the timer. You have 7 mins (guess) but I finish it in around 5 and you must return to her before the time expires and that completes the quest. If in a group swap lead (wait for respawns/repair) and let them start the quest and "chat" before killing anything.
Took a few tries to get the sequence but fully repeatable.
1) There is no on screen timer but once you "chat" with her after you have the quest she give a response that basically says Start/Go.
2) I have completed this quest in a party and solo.
3) Party kills do count toward the person doing the quest.
4) Killing anything prior to starting the timer ("Chat") fails the quest not matter how fast you do it.