Fujita's conjecture A_n on Kähler compactifications of contractible manifolds

Let $U$) be an $n$-dimensional contractible complex manifold and let $(M,D)$ be a Kähler smooth compactification. Fujita's conjecture $A_n$. Then

$$U\cong\mathbb{C}^n$$

and $(M,D)$ is the standard example $(\mathbb{P}^n,\mathbb{P}^{n-1})$. Fujita introduced this as the weakest of three related conjectures concerning contractible complex manifolds, homology cells, and projective space; the surrounding discussion describes the problem as wide open in dimensions $n\geq 3$.