Eliahou–Kryuchkov lift conjecture for signed associahedra

Let $\mathcal{A}_{n-2}$ be the associahedron whose vertices are $n$-dimensional descendant binary trees and whose edges are reassociation moves. Let $\mathcal{A}_{n-2}^{s}$ be the analogous graph for signed trees and signed reassociation moves, and let a path in $\mathcal{A}_{n-2}$ lift when 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 signed graph projects naturally onto the ordinary associahedron, but not every path lifts. The conjecture asserts that every pair of ordinary vertices can nevertheless be joined by at least one liftable path; no resolution is supplied in the source.