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We give a decomposition of the equivariant Kasparov category for discretequantum group with torsions. As an outcome, we show that the crossed product bya discrete quantum group in a certain class preserves the UCT. We then showthat quasidiagonality of a reduced C*-algebra of a countable discrete quantumgroup $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work ofTikuisis, White and Winter, and the results in the first part of the paper, theconverse (i.e. the quantum Rosenberg Conjecture) for a large class of countablediscrete unimodular quantum groups. We also note that the unimodularity is anecessary condition.

On the Baum--Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg Conjecture