Fluctuations and limiting distribution of L_{n,k} in the wreath product action

Determine the fluctuations and the limiting distribution of L_{n,k}, the length of the longest increasing subsequence of a uniformly random element of the wreath product S_k^n ⋊ S_n acting on {1,2,...,nk}, including identifying appropriate centering and scaling and the limiting law as n and k grow.

Background

For S_n, the Baik–Deift–Johansson theorem identifies Tracy–Widom fluctuations for the longest increasing subsequence. Analogous results exist for colored permutations and signed permutations, but the present work only establishes the mean behavior under certain growth conditions on k.

This open question asks for a full fluctuation theory for L_{n,k} in the wreath product setting, including possible universality classes and limiting distributions.