  Let $R = K[X_1, ..., X_n]$ be a polynomial ring over some field $K$. In this
paper, we prove that the $k$-th syzygy module of the residue class field $K$ of
$R$ has Stanley depth $n-1$ for $\lfloor n/2 \rfloor \leq k < n$, as it had
been conjectured by Bruns et. al. in 2010. In particular, this gives the
Stanley depth for a whole family of modules whose graded components have
dimension greater than $1$. So far, the Stanley depth is known only for a few
examples of this type. Our proof consists in a close analysis of a matching in
the Boolean algebra.
