Integrality conjecture for the Benson–Symonds invariant of symmetric-group permutation modules

Let $n$ be a positive integer, let $\lambda$ be a partition of $n$, and let $M^\lambda$ be the corresponding permutation module of the symmetric group $\mathcal{S}_n$. The Benson–Symonds invariant of $M^\lambda$ for $\mathcal{S}_n$ is denoted by $\gamma_{\mathcal{S}_n}(M^\lambda)$. Integrality conjecture. The number

$$\gamma_{\mathcal{S}_n}(M^\lambda)$$

is an integer. This conjecture is motivated by the behavior of permutation modules upon restriction to elementary abelian $p$-subgroups and by computations using the computer algebra system Magma; its resolution is not indicated in the source.