Fujita's conjecture on Fano manifolds with the cohomology of projective space

Let $M$ be an $n$-dimensional Fano manifold whose integral cohomology ring is isomorphic to that of projective space:

$$H^{\ast}(M;\mathbb{Z})\cong H^{\ast}(\mathbb{P}^n;\mathbb{Z}).$$

Fujita's cohomological characterization conjecture. Then $M\cong\mathbb{P}^n$. This is the strongest of the three related conjectures introduced by Fujita. The source gives no evidence that it has been resolved, so its status is open.