The Schur-positivity order on skew shapes is denoted by B<A if the difference of their respective Schur functions is a positive linear combination of Schur functions. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. In this talk we see that to determine the maximal connected skew shapes in the Schur-positivity order it is enough to consider a special class of ribbon shapes. We also explicitly determine the support for these ribbon shapes. This is joint work with Peter McNamara.