A negatively pinched Kähler threefold without bounded holomorphic coordinates

We construct a contractible domain in complex dimension three with a complete Kähler metric whose real sectional curvatures lie between two finite negative constants. It admits no bounded holomorphic map to ℂ${}^3$ with nowhere-vanishing Jacobian and is therefore not biholomorphic to a bounded domain. This gives a negative answer to the negatively pinched Kähler uniformization question.