A topological group $G$ is called
$\bullet$ minimal if it admits no strictly weaker Hausdorff group topology;
$\bullet$ completely minimal if it is Raikov-complete in each weaker Hausdorff group topology.
It is easy to see that each Raikov-complete minimal topological group is completely minimal and each completely minimal topological group is Raikov-complete.
Problem. Is each completely minimal topological group minimal? Equivalently, is an topological group completely minimal if and only if it is minimal and Raikov-complete?
Remark. The answer to this problem is affirmative for $\omega$-narrow topological groups of countable pseudocharacter and also for Abelian topological groups.