Plancherel dimension-ratio decay conjecture for subpartitions

Let $\lambda\vdash n$. Write $\mu\subseteq\lambda$ when $\mu$ is a partition obtained from $\lambda$ by removing boxes, and let $P_{PL}^n$ denote Plancherel measure on partitions of $n$. For a nonnegative integer $s$, consider subpartitions of size $|\lambda|-s$.

Plancherel dimension-ratio decay conjecture. There exists $\alpha>0$ such that, for all $s$,

$$P_{PL}^n\left(\left\{\lambda:\max_{\substack{\mu\colon\mu\subseteq\lambda\\|\mu|=|\lambda|-s}}\frac{\dim\mu}{\dim\lambda}>n^{-\alpha s}\right\}\right)\to 0.$$

This conjecture asserts polynomial decay of the largest dimension ratio after removing $s$ boxes, with probability tending to one under Plancherel measure. The paper presents this as an open estimate needed to control contributions from smaller partitions.