Su's h-positivity conjecture for truncated even-rank partition-poset homology

Let $\Pi_{2n}^e$ be the poset of partitions of $[2n]$ with an even number of blocks, and let $\Pi_{2n}^e(\bar{k})$ be the subposet obtained by selecting the top $k$ nontrivial ranks, where $1\leq k\leq n-1$.

Su's h-positivity conjecture. The action of $S_{2n}$ on the homology of $\Pi_{2n}^e(\bar{k})$ is a permutation module that can be written as a sum of induced modules whose stabilisers are Young subgroups of $S_{2n}$. Equivalently, its Frobenius characteristic is $h$-positive.

The conjecture concerns a representation-theoretic strengthening of the known permutation-module description for the full even-rank partition poset. The source poses it as an open problem to be proved using partitionings of quotient complexes.