The minimal conjecture for four-manifolds

A smooth, closed and oriented $4$-manifold is a smooth manifold of dimension four with the stated properties. A symplectic manifold is a smooth manifold equipped with a symplectic form, and a definite intersection form is an intersection form that is either positive or negative definite. A connected sum is formed by deleting balls and gluing along the resulting $3$-sphere boundaries. The minimal conjecture. Every smooth, closed and oriented $4$-manifold is the connected sum of symplectic manifolds, with both the symplectic and the opposite orientations allowed, and of some manifolds with definite intersection forms. The claim was proposed as a natural extension of the simply connected statement to non-simply-connected manifolds, but the source says that it is false and gives no replacement conjecture.