Conjecture on Stein universal covers and deformation invariance for solvmanifolds

A left-invariant complex structure is a complex structure induced by a left-invariant complex structure on the Lie group. A small deformation is a deformation sufficiently close to the original complex structure. Let $G$ be an even-dimensional simply connected unimodular solvable Lie group, and let $N$ be an even-dimensional nilmanifold.

Solvmanifold deformation conjecture. (i) Every left-invariant complex structure on $G$ is Stein, respectively biholomorphic to ${\bf C}^n$ in the nilpotent case. (ii) Every small deformation of a left-invariant complex structure on $N$ is left-invariant.

These conjectures address whether the Stein property of universal coverings and left-invariance of complex structures persist in the settings considered. The paper gives examples of non-left-invariant complex structures on a six-dimensional solvmanifold, so the broader preservation question is nontrivial.