The global-branch positivity conjecture for higher-intersection promotions

Let $(G,\mathbf{D})$ be a tangle admitting a brushing, with $\dim\Pi_G=km$ and positive intersection number $\operatorname{IN}_m(G)>0$. Let $F_{\widehat{(G,\mathbf{D})}}$ be the associated space and let $\operatorname{Gr}^{>0}_{m,n}$ denote the totally positive Grassmannian. Global-branch positivity conjecture. One can choose a brushing and signs for which the higher-intersection positivity conjecture holds, and

$$F_{\widehat{(G,\mathbf{D})}}\cap\left(\operatorname{Gr}^{>0}_{m,n}\times\operatorname{Gr}^{>0}_{m,D^{(1)}}\times\cdots\times\operatorname{Gr}^{>0}_{m,D^{(\ell)}}\right)$$

is the disjoint union of a fixed number of graphs of functions from $\operatorname{Gr}^{>0}_{m,n}$ to $\prod_{D\in\mathbf{D}}\operatorname{Gr}^{>0}_{m,D}$. This remains open beyond the illustrated cases.