  In this paper we prove that the duals of the quantum reflection groups have
the Haagerup property for all $N\ge4$ and $s\in[1,\infty)$. We use the
canonical arrow onto the quantum permutation groups, and we describe how the
characters of the quantum reflection groups behave with respect to this
canonical morphism thanks to the description of the fusion rules binding
irreducible corepresentations of $H_N^{s+}$. This allows us to construct states
on the central algebra of $H_{N}^{s+}$ and to use a fundamental theorem proved
by M.Brannan giving a method to construct nets of trace-preserving, normal,
unital and completely positive maps on the von Neumann algebra of a compact
quantum group of Kac type.
