We investigate the equilibrium morphological properties of a symmetric ABC triblock copolymer in the strong segregation regime. Although a tricontinuous double diamond phase has been experimentally reported within a large region of the phase diagram, we find that neither gyroid nor double diamond cubic phases are stable relative to lamellae and cylinders in the asymptotic limit of strong segregation(SSL). However, this finding does not rule out stable equilibrium cubic phases at intermediate molecular weights or moderate temperatures. Moreover, in contrast to the experimental conclusions, the gyroid structure is shown to be marginally stable with respect to the double diamond structure in the SSL.