**Questions concerning coamenability**

Here are few more open questions :

a) Reiji Tomatsu stated in

Reiji Tomatsu, Amenable discrete quantum groups, Journal of the Mathematical Society of Japan
Vol. 58 (2006) No. 4 P 949-964, see also http://arxiv.org/abs/math/0302222

that nuclear compact Kac algebras (=compact quantum groups of Kac type, i.e. with a tracial Haar state) are coamenable, and gave as on open problem, if this true more generally for all compact quantum groups?

Since Doplicher, Longo, Roberts, and Zsido (Reviews in Mathematical Physics, Vol. 14, Nos. 7 & 8 (2002) 787–796) have shown that coamenability implies nuclearity, the question is whether nuclearity and coamenability are equivalent for compact quantum groups?

Does anybody know if there has been progress on this question since 2006?

b) Under what conditions (like compact, discrete, Kac) is it true that a locally compact quantum group $G$ is amenable if and only if it's dual locally compact quantum group $\hat{G}$ is coamenable? Can anybody give references that contain the proofs for the new results? Many thanks in advance!

In which cases is the question still open?