The positive geometry conjecture for amplituhedron tiles

Let $Z_S$ be a tile of the amplituhedron $\mathcal{A}_{n,k,m}(Z)$, and let $\tilde{\Omega}(Z_S)$ denote its candidate canonical form, defined by pulling back the canonical form of the corresponding positroid cell. Tiles are positive geometries. The pair

$$(\operatorname{Gr}_{k,k+m}(\mathbb{C}),Z_S)$$

is a positive geometry, and its canonical form satisfies

$$\Omega(\operatorname{Gr}_{k,k+m}(\mathbb{C}),Z_S)=\tilde{\Omega}(Z_S).$$

This conjecture would justify constructing amplituhedron canonical forms from tile forms, but the source supplies no resolution.