Fine's positive definite connection conjecture

Let $M$ be a closed $4$-manifold. A connection on an associated sphere bundle is called positive definite when its induced symplectic structure has fibre normal degree $+2$. Fine's positive definite connection conjecture. If $M$ admits a positive definite connection, then $M$ is diffeomorphic to either $S^4$ or $\overline{\mathbb{C\mathbb{P}}}^{2}$. The conjecture first appeared in Fine's work and remains completely open; it concerns the classification of four-manifolds carrying such connections.