The re-embedding conjecture for non-simplicial MaxDeg planar order ideals

Let ${\mathcal{O}}$ be a non-simplicial MaxDeg planar order ideal, and let $\mathbb{B}_{\mathcal{O}}$ denote its border basis scheme. A $Z$-separating re-embedding means a re-embedding obtained by eliminating variables associated with a $Z$-set, and let $\mu=\#{\mathcal{O}}$. The segmentation type of ${\mathcal{O}}$ is the number of segments in its relevant border decomposition. The conjecture asserts that the following conditions are equivalent:

1. There exists a $Z$-separating re-embedding of $\mathbb{B}_{\mathcal{O}}$ which yields an isomorphism

$$\mathbb{B}_{\mathcal{O}} \cong \mathbb{A}^{2\mu}.$$

2. The segmentation type of ${\mathcal{O}}$ is one.

This conjecture predicts that, for non-simplicial MaxDeg planar order ideals, an affine-space re-embedding of the expected dimension exists exactly when the segmentation type is one. The source reports verification for a large number of explicitly computed examples, while the general equivalence remains open.