Hsiang–Kleiner diffeomorphism conjecture for positively curved 4-manifolds

Let $M$ be an orientable closed positively curved $4$-manifold with an effective isometric $S^1$-action. Hsiang–Kleiner's conjecture. $M$ is diffeomorphic to $S^4$ or ${\bf CP}^2$. This conjecture asks whether the known topological classification can be strengthened to a diffeomorphism classification. It is stated here as a question posed by Hsiang and Kleiner; the supplied text does not provide resolution evidence.