Existence of a counterexample separating universal lifts from naive lifts of cocycles

Determine whether there exists a dual unitary 2-cocycle \hat{\Omega} on a locally compact quantum group G for which a unitary 2-cocycle \hat{\Omega}_u satisfies (\hat{\pi}_{red} \otimes \hat{\pi}_{red})(\hat{\Omega}_u) = \hat{\Omega}, yet twisting the amplified comultiplication on M2(C^*(G)) by \hat{\Omega}_u fails to produce a copy of (C^*(G), \hat{\Delta}_u); equivalently, construct a concrete example demonstrating that admitting a universal lift (as defined via a trivial universal linking C^*-algebra) is strictly stronger than the existence of such \hat{\Omega}_u, or prove these notions coincide.

Background

The authors introduce the notion of a unitary 2-cocycle admitting a universal lift, tied to having a trivial universal linking C*-algebra. They note that this requirement is, in principle, stronger than simply having a 2-cocycle \hat{\Omega}_u that lifts \hat{\Omega} under the reducing map.

However, they do not know a concrete example that separates these notions: i.e., a case where a naive lift exists but does not yield the correct structure when twisting the amplified comultiplication on M2(C^*(G)). Resolving this would clarify the exact relationship between universal lifts and naive lifts and solidify the general framework for cocycle twisting at the universal C*-level.