Böhm–Lafuente's dynamical Alekseevskii conjecture

Let $M$ be a homogeneous space of dimension $n$, and call a Ricci flow on $M$ immortal if it exists for all positive times $t$. The universal cover of $M$ is the simply connected covering space of $M$.

Dynamical Alekseevskii conjecture. If a homogeneous space has an immortal Ricci flow, then its universal cover is diffeomorphic to $\mathbb{R}^n$.

Lafuente proved the converse implication: if the universal cover of a homogeneous space is diffeomorphic to $\mathbb{R}^n$, then its Ricci flow is immortal. The conjecture is the dynamical version of the Alekseevskii conjecture, which states that a simply connected homogeneous negative-Einstein space is diffeomorphic to $\mathbb{R}^n$; the paper's abstract says that the conjecture is proved in dimension five.