Threshold for regularity of hook shapes

Determine the threshold for the leg length \(k\), as a function of the total size \(n\), such that higher Lie characters indexed by hook shapes \((n-k,1^k)\) tend to be regular.

Background

The paper proves regularity for hook shapes with $k=o(n^{1/4})$ and proves non-regularity when $k>(2+\varepsilon)\sqrt n$ infinitely often.

The interval between these lower and upper bounds is unresolved, motivating the request for the exact threshold.