The dual Amplituhedron volume conjecture

Let ${\mathcal A}$ be an Amplituhedron embedded in an irreducible variety $X$, and let $\mathcal Y\in X$ lie away from the boundary of ${\mathcal A}$. Dual Amplituhedron volume conjecture. There exists an irreducible dual variety $X^*_{\mathcal Y}$ and a bijection between positive geometries in $X$ and positive geometries in $X^*_{\mathcal Y}$, sending ${\mathcal A}$ to a dual Amplituhedron ${\mathcal A}^*_{\mathcal Y}$, reversing inclusions and preserving triangulations. Moreover, there is a $\mathcal Y$-dependent measure $d\operatorname{Vol}$ on $X^*_{\mathcal Y}(\mathbb R)$ such that

$$\underline{\Omega}(\mathcal B)(\mathcal Y)=\operatorname{Vol}(\mathcal B^*_{\mathcal Y}):=\int_{\mathcal B^*_{\mathcal Y}}d\operatorname{Vol}$$

for every positive geometry $\mathcal B$ in $X$. The conjecture extends the dual-polytope volume formula to Amplituhedra; the source restricts the discussion to even $m$ because the volume formulation requires positive convexity, which generally fails for odd $m$.