A weaker published characterization

Thus self-duality and irreducibility are not logically necessary for cone selection once homogeneity and strict convexity are assumed. We retain the proof under $\mathsf{S}$ because it exposes the pairing assumption and the exclusion of higher-rank positive cones. The weaker theorem provides a precise alternative without invoking an unproved “homogeneous oval” argument or a gauge-Hessian calculation.