So I just beat the second Augur, and watched the little scene that played out afterwards. However, when I tried to enter the Cave of Steam, it was blocked by a character who said 'Scram kid, you're not allowed in here. The Sky Cult is busy with Helios City right now'.
I tried refreshing the map, and exiting and re-entering the area, but nothing happened.
Then I climbed back to the top of the Jade Tower, and was trapped by Damian who wouldn't let me leave the area until I had beaten the Augur. However, when I went back on the roof, the Augur wasn't there (not too surprising as I had just beaten him), and I was stuck up on the roof.
It seems as though the game hasn't acknowledged me beating the Augur.