Ball and positive-geometry conjecture for the $m=2$ amplituhedron

For ${\mathcal{N}} \in Q_{n,2}$, let $A_{\mathcal{N}}:=A_{n,k,2}\cap {\mathring{\Pi}}_{\mathcal{N}}$, where $A_{n,k,2}$ is the $m=2$ amplituhedron and ${\mathring{\Pi}}_{\mathcal{N}}$ is the corresponding positroid stratum. Let $d({\mathcal{N}})$ denote the dimension of the stratum. Ball and positive-geometry conjecture. The stratum $A_{\mathcal{N}}$ is an open ball of dimension $d({\mathcal{N}})$, and its closure $\overline{A_{\mathcal{N}}}$ is a closed ball of the same dimension. Moreover, $A_{n,k,2}$, together with the stratification $\{A_{\mathcal{N}}\}$, is a regular CW-complex homeomorphic to a closed ball of dimension $2k$, and is a positive geometry. This would give a topological and geometric description of the face stratification of the $m=2$ amplituhedron; the preceding theorem establishes the description of its nonempty open faces and their closure relations, while the ball and positive-geometry assertions remain unproved in the supplied text.