Nonexistence of positive sectional curvature on $\mathbb{S}^{2}\times\mathbb{S}^{2}$

Let $\mathbb{S}^{2}\times\mathbb{S}^{2}$ be the product of two 2-spheres. Hopf's product conjecture. There is no Riemannian metric on $\mathbb{S}^{2}\times\mathbb{S}^{2}$ with positive sectional curvature. The statement is one of the Hopf conjectures recalled after the proposed flow conjecture. The supplied text gives no evidence of resolution.