Recently, Cohen and Wales built a faithful linear representation of the Artin
group of type $D_n$, hence showing the linearity of this group. It was later
discovered that this representation is reducible for some complex values of its
two parameters. It was also shown that when the representation is reducible,
the action on a proper invariant subspace is a Hecke algebra action of type
$D_n$. The goal of this paper is to classify these proper invariant subspaces
in terms of Specht modules indexed by double partitions of the integer $n$.
This work is the continuation of arXiv:1103.5673