Harmonic maps from spheres with lowest possible index and their properties

In this note, we prove that for n > 2 a harmonic map ${\mathbb{S}}^n\to {\mathbb{C}\mathbb{P}}^m$, of index n+1 is pseudo horizontally weakly conformal, that $n$ must be odd. In particular, for even n every non-constant harmonic map ${\mathbb{S}}^n\to {\mathbb{C}\mathbb{P}}^m$ has index at least n+2. We also present a harmonic map from ${\mathbb{S}}^4$ to ${\mathrm{G}\mathrm{r}}_{\mathbb{C}}(2,4)$ of index 5 that is not pseudo horizontally weakly conformal, so the statement does not extend to Grassmannians.