Regularity for higher Lie characters with unbounded parts

Prove or disprove that for every sequence of partitions \(\mu^{(n)}\vdash n\) with no bounded parts, the higher Lie characters \(\psi^{\mu^{(n)}}\) are asymptotically norm equal to the regular character of the symmetric group.

Background

This conjecture proposes a broad extension of the paper’s asymptotic regularity results. The condition that the partitions have no bounded parts excludes persistent small row lengths, while allowing much more general shapes than the rectangular, hook, or random families treated in the main text.

The conjecture is stated in terms of asymptotic norm equality with the regular representation, a weaker global comparison than the multiplicity-wise regularity used in the principal theorems.