Signed-seed realization of general amplituhedron tiles

Establish, for every positive integer triple n,k,m with n\geq k+m and every strictly totally positive matrix Z\in\operatorname{Mat}^{>0}_{n,k+m}, that each open tile of the amplituhedron \mathcal{A}_{n,k,m}(Z) is defined by sign conditions on the variables of a seed of \operatorname{Gr}_{m,n}, and that after freezing the facet variables these data form a signed seed.

Background

The paper proposes that amplituhedron tiles should arise from signed seeds in the cluster algebra of the Grassmannian \operatorname{Gr}_{m,n}. More precisely, the conjecture requires a seed with s=m(n-m)+1 extended cluster variables whose signed positivity region is the given open tile, together with a subset of facet variables that can be frozen so that the signs satisfy the signed-seed balancing condition.

The conjecture is known in the paper for m=2, using the description of tiles by bicolored subdivisions, and the authors also note that it holds for certain m=4 BCFW tiles. The general case remains unresolved.