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

Let $(M,g)$ be a smooth four-dimensional Einstein manifold satisfying

$$\operatorname{Rc}=\lambda g$$

for a constant $\lambda>0$, and suppose that its sectional curvature is non-negative. The metrics $g_0$, $g_{FS}$, and the product metric are respectively the round metric on $\mathbb{S}^4$, the Fubini–Study metric on $\mathbb{CP}_2$, and the product of equal-curvature round metrics on $\mathbb{S}^2\times\mathbb{S}^2$. Folk conjecture. The manifold must be either $\mathbb{S}^4$, $\mathbb{CP}_2$, $\mathbb{S}^2\times\mathbb{S}^2$, or a quotient of one of these spaces. This is a classification conjecture motivated by the known irreducible and reducible symmetric examples; its resolution is not specified in the supplied source context.