Arkani-Hamed–Trnka triangulation conjecture for the amplituhedron

Let $Z\in\operatorname{Mat}_{k+4,n}^{>0}$, and let $\mathcal{C}_{n,k,4}$ be the collection of BCFW cells in $\operatorname{Gr}_{k,n}^{\geq 0}$. Arkani-Hamed–Trnka triangulation conjecture. The images under $\widetilde{Z}$ of the cells in $\mathcal{C}_{n,k,4}$ should triangulate the $m=4$ amplituhedron: they are pairwise disjoint and together cover a dense subset of $\mathcal{A}_{n,k,4}(Z)$. This conjecture concerns the geometric decomposition underlying canonical forms for scattering amplitudes in supersymmetric Yang–Mills theory. The paper proves disjointness when $k=2$, but the full triangulation statement remains open in general.