Final answer:
The statement about Polynices and Eteocles going to war over kingship is true; the conflict arises from Eteocles' refusal to honor their agreement to alternate the throne annually.
Step-by-step explanation:
The statement that Polynices and Eteocles went to war against each other because Eteocles refused to step down from his kingship is True. This conflict is central to the ancient Greek tragedy 'Antigone', which is part of Sophocles' Theban plays. The brothers agreed to alternate rule each year, but after the first year, Eteocles did not honor this agreement and retained the throne. This led Polynices to raise an army and attack the city of Thebes, which resulted in both brothers killing each other in battle, as foretold by the curse on their father, Oedipus.