Minimizing tangent maps into complex projective space

We classify all nonconstant smooth harmonic maps ${\mathbb{S}}^2\to {\mathbb{C}\mathbb{P}}^n$ and ${\mathbb{S}}^3\to {\mathbb{C}\mathbb{P}}^n$ whose 0-homogeneous extensions are minimizing: they are, respectively, isometric embeddings or holomorphic isometric embeddings composed with a Hopf map.