  Reiner, Shaw and van Willigenburg showed that if two skew Schur functions s_A
and s_B are equal, then the skew shapes A and B must have the same "row overlap
partitions." Here we show that these row overlap equalities are also implied by
a much weaker condition than skew Schur equality: that s_A and s_B have the
same support when expanded in the fundamental quasisymmetric basis F.
Surprisingly, there is significant evidence supporting a conjecture that the
converse is also true.
  In fact, we work in terms of inequalities, showing that if the F-support of
s_A contains that of s_B, then the row overlap partitions of A are dominated by
those of B, and again conjecture that the converse also holds. Our evidence in
favor of these conjectures includes their consistency with a complete
determination of all F-support containment relations for F-multiplicity-free
skew Schur functions. We conclude with a consideration of how some other
quasisymmetric bases fit into our framework.
