  In 2012 Jan Saxl conjectured that all irreducible representations of the
symmetric group occur in the decomposition of the tensor square of the
irreducible representation corresponding to the staircase partition. We make
progress on this conjecture by proving the occurrence of all those irreducibles
which correspond to partitions that are comparable to the staircase partition
in the dominance order. Moreover, we use our result to show the occurrence of
all irreducibles corresponding to hook partitions. This generalizes results by
Pak, Panova, and Vallejo from 2013.
