What is the main conflict in Number 9 Dream by David Mitchell?

Eiji Miyake and his twin sister, Anju, are the illegitimate children of an affair between their mother and a married man. Raised by their grandmother, they have little contact with their mother, and they do not know the identity of their father. Following Anju's death at the age of eleven, Eiji rejects his mother entirely and becomes obsessed with discovering the identity of his and Anju's father.