The Heaviside filtration conjecture for Orlik–Terao representations

Let $M_n$ and $D_n$ be graded representations of $S_n$, and let $\mathbb{C}[S_n/Z_n]$ carry the Heaviside filtration obtained by pushing forward the Heaviside filtration on the regular representation $\mathbb{C}[S_n]$ under averaging over the right action of the cyclic subgroup $Z_n$.

Heaviside filtration conjecture. The graded representations $M_n$ and $D_n$ of $S_n$ are both isomorphic to the associated graded representation of the Heaviside filtration on $\mathbb{C}[S_n/Z_n]$.

This strengthens the ungraded conjecture by predicting the grading, and the source reports verification at the level of graded vector spaces through $n=5$. No general resolution is stated.