Cuttler–Greene–Skandera monotonicity conjecture for normalized Schur functions

Let $\lambda$ and $\mu$ be partitions, and let $\bm{x}=(x_1,\ldots,x_n)$ have nonnegative entries. For a partition $\lambda$, let $S_\lambda(\bm{x})=s_\lambda(\bm{x})/s_\lambda(1^n)$ be its normalized Schur function, and write $\lambda\prec\mu$ when $\lambda$ is majorized by $\mu$. Cuttler–Greene–Skandera's conjecture.

$$S_\lambda(\bm{x})\leq S_\mu(\bm{x})\quad\text{if and only if}\quad\lambda\prec\mu.$$

The conjecture concerns monotonicity of normalized Schur functions under the dominance order on partitions. The paper states that it proves this conjecture.