Hasegawa's biholomorphism conjecture for nilpotent Lie groups

Let $G$ be a simply connected, nilpotent Lie group of dimension $2n$ and let $J$ be a left-invariant complex structure on $G$. Hasegawa's biholomorphism conjecture. The complex manifold $(G,J)$ is biholomorphic to $\mathbb{C}^n$. The conjecture is proved in the source, providing a uniformisation result for left-invariant complex structures on simply connected nilpotent Lie groups.