Same-singularity-type conjecture for similarly shaped partitions

Let $\lambda_1$ and $\lambda_2$ be $r$-dimensional partitions of lengths $l_1$ and $l_2$, respectively. Suppose that they have similar shapes and

$$\operatorname{ex.dim}_{I_{\lambda_1}}\operatorname{Hilb}^{l_1}(\mathbb{A}^r)=\operatorname{ex.dim}_{I_{\lambda_2}}\operatorname{Hilb}^{l_2}(\mathbb{A}^r).$$

Same-singularity-type conjecture. Then the germs

$$\operatorname{Hilb}^{l_1}(\mathbb{A}^3)_{Z_{I_{\lambda_1}}} \quad\text{and}\quad \operatorname{Hilb}^{l_2}(\mathbb{A}^3)_{Z_{I_{\lambda_2}}}$$

are singularities of the same type.

This generalizes the observed agreement between local models for similarly shaped partitions. The statement is based on examples and computations in the paper and remains open.