Actually, it was Gazerbeam who wrote Kronos on the walls of the cave, as his final action before dying.
I assume the subtext is that he figured out the evil plan and investigated it before becoming badly injured and seeking refuge in the cave.
Lawyers are smart after all and Gazerbeam was one in his civilian occupation.
Knowing that he wasn’t likely to live, he used his last moments of life to carve the password on the wall for anyone else who would fall into the same trap.
The rest of the theory is intriguing though, it might explain Edna’s comment about gods and the fresco in her home.