Higher-dimensional polygraph freeness conjecture

Let $E=\mathbb{A}^d$, let $Z(n,l)$ be the polygraph over $E$, and let $\mathbf{z}$ denote one chosen set of coordinates on $E^n$; write $\mathbf{x},\mathbf{y},\ldots,\mathbf{z}$ for the coordinate sets on $E^n$.

Higher-dimensional polygraph conjecture. The coordinate ring of $Z(n,l)$ over $E=\mathbb{A}^d$ is a free $k[\mathbf{z}]$-module.

The conjecture is presented as a higher-dimensional version of the polygraph theorem. It would imply that the higher-dimensional isospectral Hilbert scheme is a blowup, arithmetically normal in its blowup embedding, and flat over the coordinate space in any one set of variables.