Zoll Manifolds of Type ${\mathbb{C}\mathbb{P}}^n$ with Entire Grauert Tubes

We show that a Zoll manifold of type ${\mathbb{C}\mathbb{P}}^n$ with an entire Grauert tube is isometric to ${\mathbb{C}\mathbb{P}}^n$ with the canonical Fubini-Study metric, up to constant multiplication.