The ordered-partition Hilbert-series formula for the super-diagonal coinvariant ring

Let $n\geq1$. For an ordered set partition $\Pi$ of $[n]$, let $|\Pi|$ be its number of blocks, let $\tau(\Pi)$ be its associated permutation, and let $w_\Pi(c)$ be the schedule number of $c$. Let $[m]_q=1+q+\cdots+q^{m-1}$. The ordered-partition Hilbert-series conjecture.

$$\operatorname{Hilb}(\operatorname{SDR}_n;q,t)=\sum_{\Pi}z^{n-|\Pi|}t^{\operatorname{maj}(\tau(\Pi))}\prod_{c=1}^{n}[w_\Pi(c)]_q,$$

where the sum ranges over all ordered set partitions of $[n]$. This is proposed as a compact combinatorial expression for the Hilbert series of the super-diagonal coinvariant ring, but no resolution is supplied in the text.