Fujita's conjecture B_n on projective manifolds with homology-cell divisors

Let $M$ be an $n$-dimensional projective manifold and let $D$ be a smooth ample divisor on $M$. Suppose that the natural homomorphism

$$H_k(D;\mathbb{Z})\longrightarrow H_k(M;\mathbb{Z})$$

is bijective for $0\leq k\leq 2(n-1)$. Fujita's conjecture $B_n$. Then $M\cong\mathbb{P}^n$ and $D$ is a hyperplane on it. This is the intermediate member of Fujita's three related conjectures and concerns complements whose homology is that of a cell; the source supplies no resolution, so the conjecture remains open.