Yes, yes it would. I believe that is what TNT is counting on. Once it is known that every fraction will give you a secret avatar, once they win the obelisk. I believe the hardcore Avatar Hunters will be very interested in working together to get their missing avatars.
But mind you, the avatars are just mere speculations. There's no way to be sure until the next victor is known.
And I don't see why the Sway should be ruled out entirely because they won the first fight.
Rather Reliefs, statues and other things of fertility gods. Maybe even the very first known sex manual in neopian history.