I have just completed a Master Empire Hunt, in a fireteam of 3, and while my teammates incurred deaths, I did not. However, after completing the event, we all had the "Master of the Hunt" triumph complete and got our emblems, however I did not get the "Flawless Master of the Hunt" triumph. I would link the PCGR but apparently empire hunts don't have one? Apparently another user is also experiencing this: https://www.bungie.net/en/Forums/Post/257670563 The triumph is worded such that it doesn't imply running solo, and that would also be the same as the Flawless Nightmare Hunt from Shadowkeep.