An De Rijdt describes another open problem in her thesis, see the summary of Chapter 3 on page 2: The study of ergodic action of compact quantum groups. Here we have the results of Wassermann for compact groups saying, e.g., that $SU(2)$ admits ergodic action only on von Neumann algebras of finite type I. It is open if this is also true for, say, $SU(3)$. There is work by Boca, Landstad, Tomatsu, An De Rijdt for compact quantum groups.