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Woronowicz shows that the C*-algebras of quantum $SU(2)$ are isomorphic (only as C*-algebras, forgetting the quantum group structure). Are there similar results for quantum $SU(n)$ for $n \geq 3$?

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For SU(3), there's [1]. The general case is not fully worked out yet, I guess.

[1] Nagy, Gabriel; A rigidity property for quantum SU(3) groups. Advances in geometry, 297–336, Progr. Math., 172, Birkhäuser Boston, Boston, MA, 1999.

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