In the case that your party does not get quest kill credit for the first task, it's possible that this is due to the auto-quest sharing pop-up that activates as soon as one player talks to the NPC.
On my run that completed the quest for the entire group, I asked them all to decline the automatic quest pop-up and just talk to the NPC instead. (This should give every player the quest anyway. If not, have them share from their quest log.)
Hopefully this works for everyone, and possibly any other similar dungeon quest that causes an issue. This was after making several attempts to reset and accept the quest for the whole group, and meeting another group a week or two later who had just had the same situation.
Good luck.