Maybe not like that, but more like what they have done for story passes such as the previous one where you select what type of avi you would like to receive first from the avi boxes. You can choose each hunt which one for those that sometimes prefer the feminine avi over the male and vice versa.
I only received three out of the four avatars out of the avatar boxes. The one I don't have is the one I wanted the most. As a casual player, and spending my money on bills, this game is getting harder to play.