The local reversal edge conjecture for characteristic imset polytopes

Let $\mathcal{G}$ be a directed acyclic graph with skeleton $G$, and let $i$ be a vertex. Define $\mathcal{G}_{\downarrow i}$ by orienting every edge between $i$ and a neighbor toward $i$, while leaving all other edges as in $\mathcal{G}$. Write $c_\mathcal{G}$ and $c_{\mathcal{G}_{\downarrow i}}$ for their characteristic imsets, and let $\CIM_G$ denote the characteristic imset polytope associated with $G$. Local reversal edge conjecture. If $\mathcal{G}$ and $\mathcal{G}_{\downarrow i}$ are not Markov equivalent, then

$$\operatorname{conv}\left(c_\mathcal{G},c_{\mathcal{G}_{\downarrow i}}\right)$$

is an edge of $\CIM_G$. A linear bound on the diameter of $\CIM_G$ would follow from this conjecture; the source establishes quadratic bounds in general and linear bounds for trees and for the whole polytope, but does not resolve the conjecture for arbitrary skeletons.