Let $\cla$ be the $C^*$-algebra associated with $SU_q(2)$,
$J$ be the modular conjugation coming from the Haar state
and let $D$ be the generic equivariant Dirac operator for $SU_q(2)$.
We prove in this article that there is no element in $J\cla J$,
other than the scalars, that have bounded commutator with $D$.
This shows in particular that $J\cla J$ does not contain
any Poincar\'e dual for $SU_q(2)$.