Positive tropical Grassmannian fan conjecture for amplituhedron subdivisions

Let $\operatorname{Trop}^+Gr_{k+1,n}$ be the positive tropical Grassmannian, whose cones carry their natural partial order. Positive tropical Grassmannian fan conjecture. The regular positroidal subdivisions of $\mathcal{A}_{n,k,2}$ are parametrized by the cones of $\operatorname{Trop}^+Gr_{k+1,n}$, with the natural partial order on cones reflecting the refinement order on positroidal subdivisions. This predicts that the positive tropical Grassmannian acts as the secondary fan for these amplituhedron subdivisions.