The tiling-independence conjecture for amplituhedron canonical forms

Let $\mathcal{A}_{n,k,m}(Z)$ be an amplituhedron and let $\{Z_S\}_{S\in\mathcal{C}}$ be a tiling of it. Canonical form from tilings. The canonical form of the amplituhedron is obtained by summing the canonical forms of the tiles:

$$\Omega(\operatorname{Gr}_{k,k+m}(\mathbb{C}),\mathcal{A}_{n,k,m}(Z)) =\sum_{S\in\mathcal{C}}\Omega(\operatorname{Gr}_{k,k+m}(\mathbb{C}),Z_S).$$

In particular, this sum is independent of the tiling. The claim would make the canonical form computable from any tiling; the source supplies no resolution.