The proper-action characterization of algebraic quantum groups

Let $\mathbb{G}$ be a locally compact quantum group. A discrete quantum space is a pair $(M,\operatorname{\partial})$ as in the paper, and an action is proper and $\operatorname{\partial}$-preserving when it satisfies the stated properness condition and preserves $\operatorname{\partial}$. An algebraic quantum group is understood in the sense of Van Daele and Kustermans–Van Daele.

Proper-action characterization. The following are equivalent:

1. There exists a discrete quantum space $(M,\operatorname{\partial})$ and a proper $\operatorname{\partial}$-preserving right action $\alpha$ of $\mathbb{G}$ on $M$.
2. There exists a strongly dense $*$-subalgebra $\mathcal{O}(\mathbb{G}) \subset L^{\infty}(\mathbb{G})$ lying in the domain of the Haar weights, such that restricting the coproduct to $\mathcal{O}(\mathbb{G})$ yields an algebraic quantum group.

For classical locally compact groups, the analogous equivalence follows from the existence of a compact open subgroup and its proper action on the corresponding coset space. The conjecture asks whether this characterization extends to all locally compact quantum groups; the supplied text does not state a resolution.