Taubes' decomposition conjecture for smooth oriented four-manifolds

A smooth, compact, oriented $4$-manifold without boundary is understood as a connected sum of symplectic manifolds, allowing either the symplectic or the opposite orientation, and of manifolds with definite intersection form. Taubes' decomposition conjecture. Every smooth, compact, oriented $4$-manifold without boundary is a connected sum of symplectic manifolds, with either the symplectic or the opposite orientation, and of manifolds with definite intersection form. This would give a broad decomposition principle for four-manifolds based on symplectic pieces and definite intersection forms. The conjecture is false, as proved in; the source gives no further details of the counterexamples.