Donaldson's conjecture on stabilization of symplectic four-manifolds

Let $(X,\omega_X)$ and $(Y,\omega_Y)$ be symplectic four-manifolds that are homeomorphic. Write $\omega_{st}$ for the standard Kähler form on projective space, and identify their stabilized products by a diffeomorphism

$$X\times\mathbb{S}^2\cong_{\mathrm{diff}}Y\times\mathbb{S}^2.$$

Two symplectic forms are equivalent if they are related by a diffeomorphism and deformation through symplectic forms. Donaldson's conjecture. The manifolds $X$ and $Y$ are diffeomorphic if and only if the product symplectic forms

$$\omega_X\oplus\omega_{st}\quad\text{and}\quad\omega_Y\oplus\omega_{st}$$

are equivalent on $X\times\mathbb{S}^2\cong_{\mathrm{diff}}Y\times\mathbb{S}^2$. The conjecture proposes a relationship between smooth equivalence of symplectic four-manifolds and equivalence of their stabilized symplectic forms; the supplied text gives no resolution, so its status remains open.