The positive definite connection deformation conjecture

Let $A$ be a positive definite connection over a compact 4-manifold, and let $S$ be the volume functional on such connections. Positive definite connection deformation conjecture. $A$ can be smoothly deformed through positive definite connections to a critical point of $S$. Such a deformation result would, together with the theorem identifying critical positive definite connections and Moser's local rigidity of symplectic structures, imply the positive definite connection sphere conjecture; it remains open.