Cell-boundary conjecture for T-duality

Let $S_{\pi}$ be a loopless $(n-c)$-dimensional cell of $Gr^+_{k+1,n}$ with $c$ connected components, where $c$ is a positive integer. Then $S_{\hat{\pi}}$ is a coloopless $(2k+1-c)$-dimensional cell of $Gr^+_{k,n}$ on which $\widetilde{Z}$ is injective. Cell-boundary conjecture. Moreover, all codimension-one boundaries of generalized triangles arise in this way. This extends the established $c=1$ case.