Cluster-adjacency conjecture for facets of amplituhedron positroid tiles

Let $Z_{\pi}$ be a positroid tile of the amplituhedron $\mathcal{A}_{n,k,m}(Z)$. A facet of $Z_{\pi}$ is a codimension-one boundary positroid tile, and suppose such a facet lies on a hypersurface defined by $Q(\langle YZ_I\rangle)=0$, where $Q$ is a polynomial in the Plücker coordinates $p_I$ on $\operatorname{Gr}_{m,n}$. Define

$$\operatorname{Facet}(Z_{\pi})=\{Q(p_I)\mid \text{a facet of }Z_{\pi}\text{ lies on }Q(\langle YZ_I\rangle)=0\}.$$

Cluster-adjacency conjecture. Each element of $\operatorname{Facet}(Z_{\pi})$ is a cluster variable for $\operatorname{Gr}_{m,n}$, and these cluster variables are pairwise compatible. Moreover, if $\widetilde Q$ is a cluster variable compatible with $\operatorname{Facet}(Z_{\pi})$, then $\widetilde Q(\langle YZ_I\rangle)$ has a fixed sign on the interior $Z^{\circ}_{\pi}$ of the tile. This generalizes the established $m=2$ cluster-adjacency theorem and remains conjectural for $m>2$.