Oriented Zoll families of minimal hypersurfaces in $\mathbb{CP}^n$ and $\mathbb{HP}^n$

Let $M$ be either $\mathbb{CP}^n$ or $\mathbb{HP}^n$. We prove that the family $\overline{\mathcal{F}}=\{Σ_σ\}_{σ\in M}$ of minimal geodesic spheres in $M$ characterizes the Fubini-Study metric $\overline{g}$, that is, $\overline{g}$ is the only metric that makes these real hypersurfaces minimal. Th