TITLE: Open problems in the theory of compact quantum groups
QUESTION [13 upvotes]: What are the important open problems in the theory of compact quantum groups? Or conjectures?
Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules of $SO(3)$ of the the form $A_{aut}(B,\varphi)$ for a finite dimensional $C^*$-algebra $B$ and a $\delta$-form $\varphi$ on $B$?
See page 83 at the end of Chapter 3 in An De Rijdt's Ph.D. thesis, available online from https://perswww.kuleuven.be/~u0018768/students/derijdt-phd-thesis.pdf

REPLY [7 votes]: I'm late to the party! Here are some I momentarily mused on. Sorry I couldn't find explicit literature stating these as open problems, but I think these are natural developments to the famous results.

Determine the von Neumann algebraic type (and the factoriality) of $A_o(F)$. Is anything known beyond the results of Vaes-Vergnioux (Duke Math. J., 2007)?
Prove the strong Baum-Connes conjecture a la Meyer-Nest for the (edit: discrete dual of) q-deformation of compact simply connected simple group $G$, beyond $SU_q(2)$ due to Voigt (Adv. Math. 2011). Even at the 'classical' limit, it seems to require the understanding of the $G_{\mathbb C}$-equivariant KK-theory of $G_{\mathbb C}/B$, which is hard as far as I understand.