The plane-partition cell-count conjecture for amplituhedron triangulations

Let $M(a,b,c)$ denote the number of plane partitions fitting in an $a\times b\times c$ box, namely

$$M(a,b,c):=\prod_{p=1}^a\prod_{q=1}^b\prod_{r=1}^c \frac{p+q+r-1}{p+q+r-2}.$$

Let $Z\in\operatorname{Gr}_{>0}(k+m,n)$ and let $f_1,\dots,f_N\in\tilde{S_n}(-k,n-k)$ be a triangulation of $\mathcal{A}_{n,k,m}(Z)$, with $m$ even. Plane-partition cell-count conjecture. The number of cells is

$$N=M(k,\ell,m/2).$$

This generalizes the conjectural cell count attributed to the BCFW and related triangulations. The source gives no resolution.