The three-bond-dimension reduction for the bridge-graph conjecture

Let $(a,b)$ and $(a',b')$ lie in $\mathbf{U}_3\cup\mathbf{V}_3\cup\mathbf{W}_3$, and consider the bridge graph with bond dimension $3$ and boundary dimensions $a,b,b',a'$. Three-bond-dimension reduction conjecture. One has

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

The paper proves a reduction showing that the preceding general bridge-graph conjecture is equivalent to this specialization at $w=3$. Thus this displayed statement is the remaining three-bond-dimension case rather than an independent stronger assertion; its resolution would settle the general conjecture through the reduction.