Polyhedrality conjecture for linkage matching fields

Let a left linkage matching field be a linkage matching field with the left-side convention, and let it satisfy the compatible right submatching property when the following holds. Let $\mu_1,\ldots,\mu_r$ be submatchings of matchings in the field on $J\sqcup I_1,\ldots,J\sqcup I_r$, where

$$I:=\bigcup_{s\in [r]} I_s\subseteq R,$$

with $|J|+1=|I|$. Then

$$T=\bigcup_{s\in [r]}\mu_s$$

is a forest on $J\sqcup I$, and each matching $\mu$ of size $|J|$ in $T$ is compatible with the matchings in the field.

Polyhedrality conjecture. A left linkage matching field is polyhedral if and only if it satisfies the compatible right submatching property.

This conjecture characterizes the linkage matching fields arising from matching ensembles, or equivalently the polyhedral ones, through an intrinsic compatibility condition on right submatchings. The source gives no resolution or status evidence beyond presenting it as a conjecture.