The character bound conjecture for high-dimensional irreducible representations of symmetric groups

Let $S_n$ be the symmetric group. An irreducible representation (irrep) $\lambda$ of $S_n$ has dimension $d_\lambda$ and character $\chi_\lambda$. For a permutation $\pi\in S_n$, let $s(\pi)$ be its support, let $c(\pi)$ be the number of its nontrivial cycles, and let $t(\pi)=s(\pi)-c(\pi)$ be the minimum number of transpositions whose product is $\pi$. Call $\lambda$ big if

$$d_\lambda>e^{-\sqrt{n}\log n}\sqrt{n!}.$$

Character bound conjecture. There is a constant $A$ such that, for sufficiently large $n$, every big $\lambda$ satisfies

$$\left|\frac{\chi_\lambda(\pi)}{d_\lambda}\right|\le A^{t(\pi)}n^{-t(\pi)/2}$$

for all $\pi\in S_n$.

This conjectured uniform estimate for normalized characters of sufficiently high-dimensional irreducible representations would imply the required smoothness bounds for typical representations and support the analysis of quantum sieve algorithms for Graph Isomorphism. Its status is not resolved in the supplied source.