The dominant solvable plabic tangle quasi-cluster conjecture

Let $(G,\mathbf{D})$ be a dominant solvable plabic tangle, where $G$ has $n$ outer boundary vertices and inner disks $D^{(1)},\dots,D^{(\ell)}$, and let $m$-VRCs denote the associated $m$-vector-relation configurations. Dominant solvable tangle conjecture. There is a normalization of the vectors in the $m$-VRCs such that geometric promotion

$$\psi:\operatorname{Gr}_{m,n}\dashrightarrow\operatorname{Gr}_{m,D^{(1)}}\times\cdots\times\operatorname{Gr}_{m,D^{(\ell)}}$$

sends totally positive elements to totally positive elements, and algebraic promotion

$$\Psi=\psi^*:\mathbb{C}(\operatorname{Gr}_{m,D^{(1)}})\otimes\cdots\otimes\mathbb{C}(\operatorname{Gr}_{m,D^{(\ell)}})\to\mathbb{C}(\operatorname{Gr}_{m,n})$$

is a quasi-cluster homomorphism after freezing some variables on the right-hand side. The paper proves the assertion for several classes, but the general statement remains open.