The orthogonal web-flow conjecture for weighted webs

Let $\Gamma=(D,A,B,w)$ be a weighted web. A current is a function $f$ on the edges of $D$ satisfying the capacity and flow inequalities at every vertex, with no flow entering $A$ and no flow leaving $B$. A current satisfying Kirchhoff's law at every vertex outside $A\cup B$ is a web-flow. An $A$-$B$ separating set is a set of vertices meeting every path from $A$ to $B$. Such a set is orthogonal to a web-flow $f$ when every vertex in the set is saturated by $f$ and $f$ has no edge from outside the roofed region's nonessential part into that part.

Orthogonal web-flow conjecture. In every weighted web there exists a web-flow $f$ and an $A$-$B$ separating set orthogonal to $f$.

This is presented as an equivalent formulation of the orthogonal-pair conjecture in weighted-web terminology. Since the paper proves the underlying countable-network theorem, this formulation is solved in the stated setting.