A conformal Gauss map does not imply a minimal surface ($A^2=\mu I$)

For a minimal surface, the shape operator has eigenvalues $k,-k$, so $⟨dN(v),dN(w)⟩=k^2⟨v,w⟩$. Conversely this metric identity only implies that the two principal curvatures have equal squares. A sphere has equal nonzero principal curvatures and satisfies the identity while having nonzero mean curvature. The distinction is between opposite and equal signs, not equality of magnitudes.