For which non-abelian locally compact groups does an analogue of the Skitovich-Darmois theorem hold, characterizing Gaussian measures via independence of two group-valued linear combinations?

The classical Skitovich-Darmois theorem characterizes Gaussian distributions on the real line by the property that two distinct linear combinations of independent random variables are themselves independent. The group analogue paper extends this to certain abelian and some structured groups, but the full picture for non-abelian locally compact groups remains deeply incomplete. The open problem is to determine precisely which non-abelian locally compact groups admit a version of this characterization, and what the correct substitute for Gaussianity should be when the group lacks commutativity.