Donaldson's positive-definite connection conjecture

Let $X$ be a compact 4-manifold admitting a positive definite connection, meaning a connection whose curvature components form an oriented frame for $\Lambda^+$. Donaldson's positive-definite connection conjecture. Then $X$ also admits an anti-self-dual Einstein metric with positive scalar curvature. In particular, by Hitchin's theorem, $X$ is diffeomorphic to $S^4$ or $\mathbb C\mathbb P^2$. The negatively curved analogue is stated to be false, whereas the positively curved version is presented as speculative; no resolution of this positive version is supplied.