The Plancherel transition-measure convergence-rate conjecture

Let $\lambda^{(n)}$ be a random Young diagram with $n$ boxes distributed according to the Plancherel measure. Let $K_{\lambda^{(n)}}$ be its transition-measure distribution function and let $F_{\operatorname{SC}}$ be the semicircle distribution function. Plancherel transition-measure convergence-rate conjecture. For every $-2<u<2$ and every $\alpha<\frac12$,

$$n^\alpha\left[K_{\lambda^{(n)}}(\sqrt n\,u)-F_{\operatorname{SC}}(u)\right]\xrightarrow[n\to\infty]{P}0.$$

This stronger conjecture is motivated by simulations and analogous random-matrix results, and is intended to supply the local fluctuation estimate needed for the proposed approach to the jeu de taquin functional central limit theorem.