Asymptotic growth rate of L_{n,k} for bounded k in the wreath product action

Determine the asymptotic growth rate 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}, in the regime where k is bounded (e.g., k fixed) as n → ∞.

Background

The paper proves that L_{n,k}/(4√(nk)) → 1 in probability when n,k → ∞ with k/(log n)^4 → ∞. However, the authors explain that extending this to bounded k is difficult due to potential anomalous increasing subsequences arising from many block-level subsequences.

This open problem seeks the correct asymptotic order of L_{n,k} when k does not grow with n, complementing the main theorem and paralleling classical results for S_n and colored permutations.