A solvmanifold with a left-invariant complex structure whose universal cover is not Stein

We construct a simply connected solvable Lie group $G$ admitting lattices and a left-invariant complex structure $J$ such that $(G,J)$ is not Stein. This provides a counterexample to Hasegawa’s conjecture on the Stein property of simply connected unimodular solvable Lie groups with left-invariant complex structures.