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 q-deformations for compact simply connected simple groups $G$, 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.