Equivariant classification conjecture for positively curved 4-manifolds with circle actions

Let $M$ be a simply connected positively curved 4-manifold equipped with an isometric circle action. A linear circle action is a circle action induced by a linear action on either the sphere or complex projective plane. Equivariant classification conjecture. The action on $M$ is equivariantly diffeomorphic to a linear circle action on either $\mathbb{S}^4$ or $\mathbb{C\mkern1mu P}^2$. This proposes a complete equivariant classification of such actions; the preceding discussion explains that the classification up to equivariant diffeomorphism is not yet known because the possible weighted orbit-space data in positive curvature are not fully understood.