The Whitehouse-module restriction conjecture for the LAnKe representations

For integers $n$ and $k$, let $\rho_{n,k}$ be the representations in the LAnKe array, and let $W_{k+1}$ denote the Whitehouse module. For each irreducible submodule of $W_{k+1}$, form a Young diagram by adding $n-2$ rows of length $k-1$ to its top; let $W_{n,k}$ be the resulting representation of $S_{kn-n-k+3}$. Whitehouse-module restriction conjecture. The representation $\rho_{n,k}$ is the restriction of $W_{n,k}$ to $S_{kn-n-k+2}$.

This conjecture generalizes the relationship between the representations in the LAnKe array and the Whitehouse module, extending the preceding Catalan and Whitehouse-module observations. Finding analogs of parts (a) and (b) of the cited theorem remains an open problem.