The bridge-graph quantum max-flow conjecture in the symmetric regions

Let $w$ be a positive integer and let $(a,b)$ and $(a',b')$ lie in the union of the regions $\mathbf{U}_w$, $\mathbf{V}_w$, and $\mathbf{W}_w$. Consider the bridge graph with bond dimension $w$ and boundary dimensions $a,b,b',a'$ as indicated. Bridge-graph quantum max-flow conjecture. One has

$$\operatorname{QMaxFlow}=\operatorname{QMinCut}=\min\{a b',a'b\}.$$

This extends the symmetric-bond-dimension characterization to the general case in which both pairs belong to the specified regions. The surrounding discussion states that the remaining cases are open; the conjecture is proposed as part of the expected general equality of quantum max-flow and quantum min-cut.