Lifting property for uncountable free-group C*-algebras

Determine whether the full group C*-algebra $C^*(G)$ has the lifting property when $G$ is an uncountable free group, including the case of an uncountable free abelian group.

Background

The unresolved nonseparable question is made concrete for full group C*-algebras. The paper specifically identifies uncountable free groups, and even uncountable free abelian groups, as cases for which the lifting property has not been established.

References

Actually, it seems to be open whether $C*(G)$ has the LP when $G$ is an uncountable free (or even free abelian !) group.

— Operator space theory from the maximal viewpoint  (2610.21929 - Pisier et al., 15 Oct 2026) in Section 5, Remark immediately following the proof of Theorem \ref{d6}