Threshold for regularity of rectangular diagrams

Determine the threshold for the height \(k\), as a function of the width \(m\), such that higher Lie characters indexed by rectangular diagrams \((m^k)\) tend to be regular.

Background

The paper proves regularity for rectangles $(m^k)$ when $m\to\infty$ and $k=o(m)$, while it shows that fixed width $m=2$ and unbounded height do not yield regularity.

These results leave undetermined the transition regime between the proven regular and non-regular families. The problem asks for the precise growth threshold relating height and width.