Almost h-positivity conjecture for rank-selected subword-order homology

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 reflection representation of $S_n$, and set $T_2(n)=\{h_\lambda: \lambda=(n-r,1^r),\ r\ge 2\}$. Almost h-positivity conjecture. The homology of any finite nonempty rank-selected subposet of subword order on $A^*$, plus or minus one copy of $S_{(n-1,1)}$, is a permutation module. Its Frobenius characteristic is $h$-positive and supported on $T_2(n)$.