Moseley–Proudfoot–Young conjecture on the cohomology of configuration spaces in $SU_2$

Let $D_n$ be the cohomology algebra of the configuration spaces of $n$ ordered points in $SU_2$ up to translations, and let $M_n$ be the intersection cohomology ring of the hypertoric variety associated with the root system of type $A_n$. Both carry graded actions of the symmetric group $S_n$. Moseley–Proudfoot–Young conjecture. For each $n$, there exists an isomorphism of graded $S_n$-representations

$$D_n \simeq M_n.$$

The conjecture predicts a representation-theoretic relation between configuration-space cohomology and hypertoric intersection cohomology. In the paper, the authors prove this conjecture, so its status is solved.