The Berry-Esseen bound for character ratios

Let $\lambda$ be a partition of $n$ chosen from the Plancherel measure of the symmetric group $S_n$, let $\chi^\lambda(12)$ be the irreducible character of the symmetric group parameterized by $\lambda$ evaluated on the transposition $(12)$, and let $\dim(\lambda)$ be the dimension of the irreducible representation parameterized by $\lambda$. Fulman recently obtained the convergence rate of $O(n^{-s})$ for any $0< s<\frac 12$ in the central limit theorem for character ratios ${(n-1) \over \sqrt{2} } \, {\chi^\lambda(12) \over \dim(\lambda)}$ by developing a connection between martingale and character ratios, and he conjectures that the correct speed is $O(n^{-1/2})$. In this paper we confirm the conjecture via a refinement of Stein's method for exchangeable pairs.