Abelianness of asymptotic-cone fundamental groups without the SOL obstruction

Determine whether, for every Lie group G without the SOL obstruction and every non-principal ultrafilter ω, the fundamental group π₁(cone[G]) is abelian.

Background

The paper proves that a real triangulable Lie group with the SOL obstruction has asymptotic-cone fundamental groups containing the fundamental group of the Hawaiian earring, and therefore containing nonabelian free groups. It consequently asks whether excluding the SOL obstruction forces the fundamental groups of all asymptotic cones to be abelian.

References

A natural question is then the following. Let $G$ be a Lie group without the SOL obstruction. Is it true that for every non-principal ultrafilter $\omega$, the fundamental group $\pi_1(\cone[G])$ is abelian?

— Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction  (2608.30767 - Velut, 31 Aug 2026) in Section 1, subsection “Open questions” (label sec:open_questions)