The definite intersection-form conjecture for positively curved 4-manifolds

Let $M$ be a compact oriented $4$-manifold with positive sectional curvature and nonvanishing second Betti number. Its intersection form is the bilinear form on $H^2(M;\mathbb{R})$ induced by the cup product and evaluation on the fundamental class. Definite intersection-form conjecture. The intersection form of $M$ is definite. This is presented as a stronger conjecture related to the Hopf conjecture, connecting positive sectional curvature with the topology of $4$-manifolds; its resolution is not supplied here.