The cluster adjacency conjecture for positroid tiles of the $m=2$ amplituhedron

Let $Z_{\hat{G}(\mathcal{T})}$ be a positroid tile of the amplituhedron $\mathcal{A}_{n,k,2}(Z)$. A facet is a codimension-one boundary piece, and each facet lies on a hypersurface $\langle Yij\rangle=0$. Via the identification with $\operatorname{Gr}_{2,n}(\mathbb{C})$, let $p_{ij}$ denote the corresponding Plücker coordinate. Cluster adjacency conjecture. The collection of Plücker coordinates $\{p_{ij}\}_{\hat{G}(\mathcal{T})}$ corresponding to the facets is a collection of compatible cluster variables for $\operatorname{Gr}_{2,n}(\mathbb{C})$. Moreover, if $p_{hl}$ is compatible with this collection, then $\langle Yhl\rangle$ has a fixed sign on the open positroid tile $Z^{\circ}_{\hat{G}(\mathcal{T})}$. The paper proves this $m=2$ conjecture, while proposing a broader version for general $m$.