Conjecture on column lengths in the decomposition of gamma modules

Let $n,k\geq 1$, and let $\gamma_{n,k}$ be an $\mathfrak S_{k(n-1)+1}$-module such that

$$\rho_{n,k}\cong\beta_{n,k}\oplus\gamma_{n,k},$$

where every irreducible constituent of $\gamma_{n,k}$ has at most $k-1$ columns. Column-length conjecture. If $n\leq j<k$, then there is an irreducible constituent of $\gamma_{n,k}$ whose Young diagram has exactly $j$ columns. This conjecture refines the converse direction of the characterization of when $\rho_{n,k}\cong\beta_{n,k}$; the supplied text gives no evidence that it has been resolved.