Eliahou–Kryuchkov path-lifting conjecture for signed associahedra

Let $\mathcal{A}_{n-2}$ be the graph whose vertices are the $n$-dimensional descendant binary trees and whose edges are reassociation moves, and let $\mathcal{A}_{n-2}^{s}$ be the analogous graph for signed trees and signed reassociation moves. A path in $\mathcal{A}_{n-2}$ is liftable if it has a corresponding path in $\mathcal{A}_{n-2}^{s}$.

Eliahou–Kryuchkov conjecture. For any pair of vertices on $\mathcal{A}_{n-2}$, there exists a path connecting them that can be lifted to a path on the graph $\mathcal{A}_{n-2}^{s}$.

The conjecture asks whether every pair of vertices in the ordinary associahedron can be joined by a path compatible with the signed-tree structure. The supplied text attributes it to Eliahou and Kryuchkov but gives no information about its resolution, so its status is left open.