Young-subgroup positivity conjecture for Boolean matroid Chow rings

Let $\mathcal{M}$ be a Boolean matroid of rank $n$, let $\mathfrak{S}_n$ act on its Chow ring, and let $\mathrm{FY}^i$ denote the associated $\mathfrak{S}_n$-set. For $\lambda=(\lambda_1,\ldots,\lambda_\ell)$, write

$$\mathfrak{S}_\lambda:=\mathfrak{S}_{\lambda_1}\times\mathfrak{S}_{\lambda_2}\times\cdots\times\mathfrak{S}_{\lambda_\ell}.$$

Young-subgroup positivity conjecture. For $i\leq j\leq k\leq\ell$ with $i+\ell=j+k$, the element

$$[\mathrm{FY}^j][\mathrm{FY}^k]-[\mathrm{FY}^i][\mathrm{FY}^\ell]\geq_{B(\mathfrak{S}_n)}0$$

is a genuine permutation representation whose orbit stabilizers are all Young subgroups $\mathfrak{S}_\lambda$. This refines Burnside log-concavity for Boolean matroids by specifying the stabilizers occurring in the resulting permutation representation. It is presented as a conjectural strengthening suggested by the Boolean examples, while Burnside $PF_3$ itself is known to fail even in rank $4$.