  Two partial orders on a reflection group, the codimension order and the
prefix order, are together called the absolute order when they agree. We show
that in this case the absolute order on a complex reflection group has the
strong Sperner property, except possibly for the Coxeter group of type $D_n$,
for which this property is conjectural. The Sperner property had previously
been established for the noncrossing partition lattice $NC_W$, a certain
maximal interval in the absolute order, but not for the entire poset, except in
the case of the symmetric group. We also show that neither the codimension
order nor the prefix order has the Sperner property for general complex
reflection groups.
