  The Schur-positivity order on skew shapes is defined by B \leq A if the
difference s_A - s_B is Schur-positive. It is an open problem to determine
those connected skew shapes that are maximal with respect to this ordering. A
strong necessary condition for the Schur-positivity of s_A - s_B is that the
support of B is contained in that of A, where the support of B is defined to be
the set of partitions lambda for which s_lambda appears in the Schur expansion
of s_B. We show that to determine the maximal connected skew shapes in the
Schur-positivity order and this support containment order, it suffices to
consider a special class of ribbon shapes. We explicitly determine the support
for these ribbon shapes, thereby determining the maximal connected skew shapes
in the support containment order.
