Moore and Russell's character bound for balanced Young diagrams

Let $C>0$. A Young diagram $\lambda$ with $n$ boxes has at most $C\sqrt{n}$ rows and columns, and let $\pi\in S_n$ be a permutation. Moore and Russell's character bound. There exists a constant $D$ such that

$$|\chi^{\lambda}(\pi)| < \left(\frac{D}{\sqrt{n}}\right)^{|\pi|}.$$

This conjectured estimate would bound the multiplicities in Kronecker tensor products of balanced irreducible representations and would imply that the relevant quantum algorithm for graph isomorphism is no faster than the best known classical algorithms. Its resolution is not specified in the source.