Converse of representation-to-action without regularity for locally compact quantum groups

Determine whether, for a locally compact quantum group \mathbb{G} not assumed regular, every nondegenerate representation of the universal C*-algebra C^*_u(\mathbb{G}) on a Hilbert B-module E gives rise to a \mathbb{G}-action on E (i.e., a coaction of C_0^r(\mathbb{G}) on E acting trivially on B), thereby establishing the converse of Lemma \ref{lem:reg-needed?} without the regularity hypothesis.

Background

Lemma \ref{lem:reg-needed?} shows that if a Hilbert B-module E carries a \mathbb{G}-action (with \mathbb{G} acting trivially on B), then C^*_u(\mathbb{G}) is represented on E. Conversely, if \mathbb{G} is regular, any nondegenerate representation of C^*_u(\mathbb{G}) on E yields a \mathbb{G}-action on E that acts trivially on B.

The authors explicitly note uncertainty about whether this converse implication continues to hold without assuming regularity. Resolving this would clarify the relationship between representations of C^*_u(\mathbb{G}) and coactions on Hilbert modules in the general (non-regular) locally compact quantum group setting.