The hypertoric intersection-cohomology conjecture for type A

Let $X_n$ be the complex affine hypertoric variety associated with the root system of $\mathfrak{sl}_n$, equipped with its $\mathfrak{S}_n$-action, and let $M_n:=I\!H^{2*}(X_n;\mathbb{Q})$. Let $D_n$ be the graded $\mathfrak{S}_n$-representation defined earlier in the paper.

The hypertoric intersection-cohomology conjecture. For all $n$, there exists an isomorphism of graded $\mathfrak{S}_n$-representations

$$D_n\cong M_n.$$

This conjecture appeared previously in the cited work of Moci, Proudfoot, and Young. The paper proves the isomorphism in degrees at most $7$, while the assertion in all degrees remains open.