Normality and Cohen–Macaulayness of higher-dimensional isospectral Hilbert schemes

For integers $d$ and $n$, let $H_n(\mathbb{C}^d)$ be the principal component of $\operatorname{Hilb}^n(\mathbb{C}^d)$, and let $X_n(\mathbb{C}^d)$ be the isospectral Hilbert scheme over it.

Normality and Cohen–Macaulayness conjecture. For all $d$ and $n$, $X_n(\mathbb{C}^d)$ is normal and Cohen–Macaulay.

The conjecture would imply the same properties for the principal component because $H_n(\mathbb{C}^d)=X_n(\mathbb{C}^d)/S_n$. The source calls it speculative, verifies the case $d=3$, $n=4$ computationally, and leaves the general assertion open.