T-duality conjecture for hypersimplex and amplituhedron dissections

Let $\{S_{\pi}\}$ be a collection of cells of $Gr^+_{k+1,n}$, and let $\{S_{\hat{\pi}}\}$ be its T-dual collection in $Gr^+_{k,n}$. T-duality conjecture. The collection $\{S_{\pi}\}$ gives a triangulation, respectively dissection, of the hypersimplex $\Delta_{k+1,n}$ if and only if $\{S_{\hat{\pi}}\}$ gives a triangulation, respectively dissection, of the amplituhedron $\mathcal{A}_{n,k,2}$. The conjecture proposes a combinatorial correspondence between these two classes of decompositions.