  McNamara and Pylyavskyy conjectured precisely which connected skew shapes are
maximal in the Schur-positivity order, which says that $B\leq _s A$ if
$s_A-s_B$ is Schur-positive. Towards this, McNamara and van Willigenburg proved
that it suffices to study equitable ribbons, namely ribbons whose row lengths
are all of length $a$ or $(a+1)$ for $a\geq 2$. In this paper we confirm the
conjecture of McNamara and Pylyavskyy in all cases where the comparable
equitable ribbons form a chain. We also confirm a conjecture of McNamara and
van Willigenburg regarding which equitable ribbons in general are minimal.
  Additionally, we establish two sufficient conditions for the difference of
two ribbons to be Schur-positive, which manifest as diagrammatic operations on
ribbons. We also deduce two necessary conditions for the difference of two
equitable ribbons to be Schur-positive that rely on rows of length $a$ being at
the end, or on rows of length $(a+1)$ being evenly distributed.
