Arkani-Hamed and Trnka's BCFW triangulation conjecture for the amplituhedron

Let $k\geq 1$ and $n\geq k+4$. Let $\mathcal{A}_{n,k,4}$ be the amplituhedron, and let $\mathcal{BCFW}_{n,k}$ denote the collection of $4k$-dimensional BCFW positroid cells in the nonnegative Grassmannian $\mathrm{Gr}^{\geq}_{k,n}$ arising from the BCFW recurrence. A triangulation is a collection of $4k$-dimensional open positroid cells whose images under the amplituhedron map are injective on each cell, pairwise disjoint, and cover an open dense subset of the amplituhedron. Arkani-Hamed and Trnka's conjecture. For every $k\geq 1$ and $n\geq k+4$, the cells $\mathcal{BCFW}_{n,k}$ form a triangulation of the amplituhedron $\mathcal{A}_{n,k,4}$. This conjecture connects the BCFW recurrence for scattering amplitudes with a geometric triangulation of the amplituhedron. The supplied text gives no resolution status.