The standard-manifold classification conjecture for nonnegatively curved simply connected 4-manifolds

Let $N$ be a closed simply connected non-negatively curved $4$-manifold. The standard manifolds in the claim are $\mathbb{S}^4$, $\mathbb{CP}^2$, $\mathbb{S}^2\times\mathbb{S}^2$, and $\mathbb{CP}^2\#\pm\mathbb{CP}^2$.

Standard-manifold classification conjecture. The manifold $N$ is diffeomorphic to one of the $5$ standard manifolds

$$\mathbb{S}^4,\quad \mathbb{CP}^2,\quad \mathbb{S}^2\times\mathbb{S}^2,\quad \mathbb{CP}^2\#\mathbb{CP}^2,\quad \text{or}\quad \mathbb{CP}^2\#-\mathbb{CP}^2.$$

This is presented as a consequence of an affirmative answer to Grove's conjecture and is stated as open in the source. It concerns the classification of closed simply connected non-negatively curved $4$-manifolds up to diffeomorphism.