One-sided negative sectional curvature and the holomorphic Liouville property

We construct, in some sufficiently large fixed finite complex dimension m, a domain in ℂ${}^m$ diffeomorphic to ${\mathbb{R}}^{2m}$ that admits a complete Kähler metric with real sectional curvature at most −1 and has only constant bounded holomorphic functions. Its sectional curvatures are unbounded below. The construction gives a negative answer to the one-sided bounded-holomorphic-function question.