Nonnegative tensor-power conjecture for homology of rank-selected subword order

Let $A$ be an alphabet of size $n\ge 2$, and let $S_n$ act on $A^*$ by permuting the alphabet. A rank-selected subposet is obtained by retaining words whose ranks belong to a finite nonempty set. Let $S_{(n-1,1)}$ be the irreducible representation indexed by the partition $(n-1,1)$. Nonnegative tensor-power conjecture. The $S_n$-action on the homology of any finite nonempty rank-selected subposet of subword order on $A^*$ is a nonnegative integer combination of positive tensor powers of $S_{(n-1,1)}$.