Classification without the topological restriction

Prove that every closed Einstein four-manifold with positive sectional curvature is homothetically isometric to the round metric on the four-sphere or to the Fubini–Study metric on the complex projective plane, without assuming the topological inequality 2χ(M)−3|τ(M)|≤4.

Background

The paper proves that a smooth, closed, oriented four-manifold carrying an Einstein metric with positive sectional curvature is homothetically isometric to the round four-sphere or to the Fubini–Study metric on ±CP² under the additional restriction 2χ(M)−3|τ(M)|≤4. The authors explicitly state that they expect this classification theorem to remain valid without that restriction. Removing the inequality would therefore establish the full classification of closed positively curved Einstein four-manifolds in dimension four.