Folklore conjecture on Einstein four-manifolds with non-negative sectional curvature

Let $M$ be a simply connected Einstein four-manifold with positive scalar curvature and non-negative sectional curvature. Folklore conjecture. The manifold must be one of the following, with the indicated metric: $\mathbb{S}^4$ with its round metric, $\mathbb{CP}^2$ with its Fubini–Study metric, or $\mathbb{S}^2\times\mathbb{S}^2$ with its product metric. This conjecture concerns the classification of positively curved Einstein four-manifolds and is presented in the source as a famous folklore conjecture. The supplied text gives no resolution, so its status remains open.