Alekseevskii's conjecture for homogeneous Einstein spaces

Let $G/H$ be a connected homogeneous space of dimension $n$. An Einstein metric has Ricci tensor proportional to the metric, and negative scalar curvature means that this proportionality constant is negative. Alekseevskii's conjecture. If $G/H$ admits an Einstein metric with negative scalar curvature, then it is diffeomorphic to $\mathbb R^n$. This is a central classification problem for non-compact homogeneous Einstein spaces. The conjecture remains open in general, although it is known in several important settings, including various low-dimensional and solvmanifold cases.