The canonical rational-function push-forward conjecture for Amplituhedra

Let ${\mathcal A}(k,n,m;l^L)$ be an Amplituhedron, let $\Delta^D$ have positive coordinates $X_1,\ldots,X_D$, and let

$$\Phi:\Delta^D\longrightarrow {\mathcal A}(k,n,m;l^L)$$

be a morphism from positive coordinates to the Amplituhedron. Denote the canonical rational function of the Amplituhedron by $\underline{\Omega}({\mathcal A})(\mathcal Y)$. Amplituhedron amplitude push-forward conjecture. For every point $\mathcal Y\in\mathcal A$, the amplitude is

$$\underline{\Omega}({\mathcal A})(\mathcal Y)=\int \frac{d^DX}{\prod_{a=1}^{D}X_a}\,\delta^D(\mathcal Y;\Phi(X)).$$

Here $\mathcal Y$ and $\Phi(X)$ are represented by matrices modulo a left group action. This is the rational-function version of the preceding form push-forward proposal, and its validity depends on constructing suitable morphisms.