Equivariant log concavity and the -module structure on $H^i(\mathrm{Conf}(n,{\mathbb{R}}^d))$

Previous work has conjectured that the graded ${\mathfrak{S}}_n$-representations $H^{\bullet }(\mathrm{Conf}(n,{\mathbb{R}}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging the theory of representation stability, we are able instead prove a stronger statement about the $\mathrm{}$-module structure on $H^i(\mathrm{Conf}(n,{\mathbb{R}}^d);\mathbb{Q})$ which implies the original conjecture up to degree 19. We conjecture that this equivariant log concavity-like property holds in all degrees for the $\mathrm{}$-modules $H^i(\mathrm{Conf}(n,{\mathbb{R}}^d);\mathbb{Q})$.