KleinerHsiang (JDG 1989) proved such a manifold is homeomorphic to $S^4$ or $CP^{2}$. an interesting corrollary is that $S^2 \times S^2$ does not admit positively curved metric with countinuous symmetry. They asked in their paper if it is diffeomorphic. I don't know much on this area, only noticed one subsequent work SearleYang. Seems that there are preprints on arXiv attempting to do the problem in full generality. Not sure if it was fully settled. Anyone knows the precise status of this problem?
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The status of the problem is that it is solved: the classification up to (equivariant) diffeomorphism was obtained very recently by K. Grove and B. Wilking, see Thm A here. The proof uses the solution of the Poincare' Conjecture and some fancy machinery of Alexandrov geometry. 

