Conjecture on column lengths in the decomposition of rho modules

For integers $n,k\geq 1$, let $\rho_{n,k}$ be the representation considered in the paper, an $\mathfrak S_{k(n-1)+1}$-module. Column-length conjecture. If $n\leq j\leq k$, then there is an irreducible constituent of $\rho_{n,k}$ whose Young diagram has exactly $j$ columns. The supplied status evidence says that this claim can be proved using the result of Kraskiewicz and Weyman given in the cited theorem, so it is no longer conjectural.