You have to be alive, if you die while your team is at the boss, you cannot get back in until they finish and exit or both die as well. In order to complete the bounty in your pursuits, you MUST open the chest that spawns after the boss is killed. Then hand in the completed bounty. The chest will also reward dreaming city gear but not powerful gear. Please tell me that this is clear enough.
The bounty is tied to opening the chest