so is $\mathcal{W}^*$. Moreover, it is clear that $\mathcal{W}^*$ is pushed onto $R$ and has the same multiplicity as $\mathcal{W}'$. However, as $j > t$, the sum of the subscripts of the edge copies in $E(\mathcal{W}^*)$ is larger than that of $E(\mathcal{W}')$, which contradicts the choice of $\mathcal{W}'$.
Hence, there exist weak linkages $\mathcal{W}'$ that are sensible, pushed onto $R$, discretely homotopic to $\mathcal{W}$, whose multiplicity is upper bounded by the multiplicity of $\mathcal{W}$, and for all edges $e_i \in E(\mathcal{W}')$ $i \ge i$. Consider the collection of all such weak linkages, and let $\mathcal{W}^*$ be the one maximizing $w(E(\mathcal{W}^*)) = \sum e \in E(\mathcal{W}^*)w(e)$ where Let us begin by defining a weight function $w: E(H) \to \mathbb{Z}$ on the parallel copies of edges in $H$ as follows.
We claim that $\mathcal{W}^*$ is canonical, i.e. for every edge $e_i \in E(\mathcal{W}^*)$ the subscript $i \ge 1$, and for every parallel edge $e_j$ where $i \le j < i$, $e_j \in E(\mathcal{W})$. The first property is ensured by the choice of $\mathcal{W}^*$. For the second property, we argue as before. Suppose not, then choose $i$ and $j$ such that $i-j$ is minimized. Then clearly $j = i-1$, since any parallel copy $e_t$ with $j < t < i$ is either in $E(\mathcal{W}^*)$ contradicting the choice of $i$, or not in $E(\mathcal{W}^*)$ contradicting the choice if $j$. Therefore, the edges $e_i$ and $e_j$ form a cycle $C$ such that the interior of $C$ contains no edge of any walk in $\mathcal{W}^*$. Let $\mathcal{W} \in \mathcal{W}^*$ be the walk containing $e_i$, and observe that the cycle move operation is applicable to $(\mathcal{W}, C)$. Let $\tilde{\mathcal{W}}$ be the result of this operation. Then observe that $w(E(\tilde{\mathcal{W}})) > w(E(\mathcal{W}^*))$, since $w(e_j) > w(e_i)$ and $E(\tilde{\mathcal{W}}) \setminus {e_j} = E(\mathcal{W}^*) \setminus {e_i}$. And, $\tilde{\mathcal{W}}$ is discretely homotopic to $\mathcal{W}'$ which is in turn discretely homotopic to $\mathcal{W}$, as before we can argue that $\tilde{\mathcal{W}}$ contradicts the choice of $\mathcal{W}^*$. Hence, $\mathcal{W}^*$ must be canonical. $\square$
In case we are interested only in extremality rather than canonicity, we can use the following lemma that does not increase potential. The proof is very similar to the proof of Lemma 8.8, except that now we can “move edges in either direction”, and hence avoid creating new crossings.
Lemma 8.9. Let $(G, S, T, g, k)$ be an instance of Planar Disjoint Paths, and $R$ be a Steiner tree. Let $\mathcal{W}$ be a weak linkage in $H$ that is sensible and pushed onto $R$, whose multiplicity is at most $2n$. Then, there exists a weak linkage $\mathcal{W}'$ that is sensible, pushed onto $R$, extremal, discretely homotopic to $\mathcal{W}$, and whose potential is upper bounded by the potential of $\mathcal{W}$.
Proof. Consider all weak linkages in $H$ that are sensible, pushed onto $R$, discretely homotopic to $\mathcal{W}$, and whose potential and multiplicity are upper bounded by the potential and multiplicity, respectively, of $\mathcal{W}$. Among these weak linkages, let $\mathcal{W}'$ be one such that the sum of the absolute values of the subscripts of the edge copies in $E(\mathcal{W}')$ is maximized. We claim that $\mathcal{W}'$ is extremal, which will prove the lemma. To this end, suppose by way of contradiction that $\mathcal{W}'$ is not extremal. Thus, there exist an edge $e_i, e_j \in \mathcal{W}'$ where $i \ge 1, j \le -1$ and $(i-1)+|j+1| \le 2n-1$. Because the multiplicity of $\mathcal{W}'$ is at most $2n$, this means that there exists an edge $e_p$ (parallel to $e_i$ and $e_j$) for some $p > i \ge 1$ that is not used by $E(\mathcal{W}')$. Let $e_t$ be the edge (parallel to $e_j$ and $e_i$) of largest subscript smaller than $p$ that is used by $E(\mathcal{W}')$. Moreover, let $C$ be the cycle (which might be the boundary of a single face) that consists of two edges: $e_p$ and $e_t$. By the choice of $e_t$, the strict interior of $C$ does not contain any edge of $E(\mathcal{W}')$. Thus, the cycle move operation is applicable to $(\mathcal{W}, C)$ where $\mathcal{W}$ is the walk in $\mathcal{W}'$ that uses $e_t$. Let $\mathcal{W}^*$ be the result of the application of this operation. Then, the only difference between $\mathcal{W}^*$ and $\mathcal{W}'$ is the replacement of $e_t$ by $e_p$.
Because $\mathcal{W}^*$ is discretely homotopic to $\mathcal{W}'$, $\mathcal{W}'$ is discretely homotopic to $\mathcal{W}$ and discrete homotopy is transitive, we derive that $\mathcal{W}^*$ is discrete. Moreover, the endpoints of the walks in $\mathcal{W}'$ were not changed when the cycle move operation was applied. Thus, because $\mathcal{W}'$ is sensible, so is $\mathcal{W}^*$. Moreover, it is clear that $\mathcal{W}^*$ is pushed onto $R$, because $\mathcal{W}^*$ and $\mathcal{W}'$ cross $R$ exactly in