  It is known that differences of symmetric functions corresponding to various
bases are nonnegative on the nonnegative orthant exactly when the partitions
defining them are comparable in dominance order. The only exception is the case
of homogeneous symmetric functions where it is only known that dominance of the
partitions implies nonnegativity of the corresponding difference of symmetric
functions. It was conjectured by Cuttler, Greene, and Skandera in 2011 that the
converse also holds, as in the cases of the monomial, elementary, power-sum,
and Schur bases. In this paper we provide a counterexample, showing that
homogeneous symmetric functions break the pattern. We use semidefinite
programming to find an explicit sums of squares decomposition of the polynomial
$H_{44} - H_{521}$ as a sum of 41 squares. This rational certificate of
nonnegativity disproves the conjecture, since a polynomial which is a sum of
squares cannot be negative, and since the partitions 44 and 521 are
incomparable in dominance order.
