Critical-locus conjecture for Borel Hilbert-scheme neighborhoods

Let $\lambda$ be a $3$-dimensional partition of $n$ such that $I_{\lambda}$ is a Borel ideal of $\Bbbk[X_1,X_2,X_3]$. Let $Z_I$ be the corresponding point of $\operatorname{Hilb}^n(\mathbb{A}^3)$, and let $A_{\lambda}$ be the coordinate ring of its Haiman neighborhood. Then there exists a regular function $F_{\lambda}$ on the tangent space of $\operatorname{Hilb}^n(\mathbb{A}^3)$ at $Z_I$ such that $\operatorname{Spec}(A_{\lambda})$ is isomorphic to the critical locus of $F_{\lambda}$.

Critical-locus conjecture. The Haiman neighborhood of every such Borel monomial ideal is a critical locus in the stated tangent space.

The paper states that this holds for $n\leq7$ and for pyramids, but it remains open in general. The conjecture would give a uniform critical-locus description of neighborhoods in the Hilbert scheme of points on $\mathbb{A}^3$.