We remark that the notation above differs from our earlier use of $\overline{\text{WindNum}}(\cdot)$ in the choice of the reference path in $\text{Ring}(S_u, S_v)$. For $\overline{\text{WindNum}}(\cdot)$, the path $\eta(u, v)$ was the reference path, whereas for $\text{WindNum}(\cdot, \text{Ring}(S_u, S_v))$, we choose $\text{path}_R(u', v')$ as the reference path.
We will now apply Lemma 7.1 in each ring of the form $\text{Ring}(S_{u_i}, S_{v_i})$ to obtain a solution of bounded winding number in that ring. Towards this, fix one pair $(u, v) := (u_i, v_i)$ for some $1 \le i \le t$. We argue that $\mathcal{C}(u, v)$ and $\text{Flow}_R(u, v)$ are suitable for the roles of $\mathcal{C}$ and $\mathcal{Q}$ in the premise of Lemma 7.1. Recall that $\text{Flow}R(u, v)$ is a collection of vertex disjoint paths in the subgraph $G{u_i v_i}$ of $G$, and assume w.l.o.g. that $\text{Flow}_R(u, v)$ uses only 2-nd copies of edges in $H$.
Observation 7.4. Let $(G, S, T, g, k)$ be a good Yes-instance of Planar Disjoint Paths. Let $R$ be a backbone Steiner tree. Let $u, v$ be a pair of near vertices in $V_{=1}(R) \cup V_{\ge 3}(R)$ such that $\text{path}_R(u, v)$ is a long degree-2 path in $R$. Then, the following hold.
(i) $\mathcal{C}(u, v)$ is an encircling tight sequence of concentric cycles in $\text{Ring}(S_u, S_v)$ where each cycle lies in $G_{u,v}$, $S_u$ is in its strict interior and $S_v$ is in its strict exterior. Further, $|\mathcal{C}(u, v)| \ge 40\alpha_{\text{sep}}(k)$ cycles.
(ii) $\text{Flow}R(u, v)$ is a maximum flow between $S_u \cap V(G)$ and $S_v \cap V(G)$ in $G{u,v}$ that is minimal with respect to $\mathcal{C}(u, v)$.
(iii) Each path $Q \in \text{Flow}_R(u, v)$ traverses $\text{Ring}(S_u, S_v)$, and $|\overline{\text{WindNum}}(\text{path}_R(u', v'), Q)| \le 1$.
Proof. The first statement directly follows from the construction of $\mathcal{C}(u, v)$ (see Lemma 6.6). Similarly, the second statement directly follows from the construction of $\text{Flow}_R(u, v)$ using $\mathcal{C}(u, v)$ (see Observation 6.8). Additionally the construction implies that each path $Q \in \text{Flow}_R(u, v)$ traverses $\text{Ring}(S_u, S_v)$. For the second part of the third statement, first note any $Q \in \text{Flow}_R(u, v)$ and $\text{path}_R(u', v')$ are two edge-disjoint traversing paths in $\text{Ring}(S_u, S_v)$. Hence, $\overline{\text{WindNum}}(\text{path}_R(u', v'), P)$ is well defined. Since for any $Q \in \text{Flow}_R(u, v)$, there are at most two edges in $\text{path}_R(u', v')$ with only one endpoint in $V(Q)$ (by Corollary 6.1, the absolute value of the signed sum of crossing between these two paths is upper-bounded by 1, i.e. $|\overline{\text{WindNum}}(Q, \text{path}_R(u', v'))| \le 1$. $\square$
Finally we are ready to prove that the existence of a solution of small winding number.
Lemma 7.3. Let $(G, S, T, g, k)$ be a good Yes-instance of Planar Disjoint Paths. Let $R$ be a backbone Steiner tree. Let ${u_1, v_1}, {u_2, v_2}, \dots, {u_t, v_t}$ be the pairs of near vertices in $V_{=1}(R) \cup V_{\ge 3}(R)$ such that $\text{path}R(u_i, v_i)$ is a long maximal degree-2 path in $R$ for all $i \in {1, 2, \dots, t}$. Let $\text{Ring}(S{u_1}, S_{v_1}), \text{Ring}(S_{u_2}, S_{v_2}), \dots, \text{Ring}(S_{u_t}, S_{v_t})$ be the corresponding rings. Then, there is a solution $\mathcal{P}^*$ to $(G, S, T, g, k)$ such that $|\overline{\text{WindNum}}(\mathcal{P}^*, \text{Ring}(S_{u_i}, S_{v_i}))| \le \alpha_{\text{winding}}(k)$ for all $i \in {1, 2, \dots, t}$, where $\alpha_{\text{winding}}(k) = 60\alpha_{\text{sep}}(k) + 11 < 300 \cdot 2^{ck}$.
Proof. Consider a solution $\mathcal{P}$ to $(G, S, T, g, k)$. Fix a pair $(u, v) := (u_i, v_i)$ for some $i \in {1, 2, \dots, t}$. Recall that $|S_u|, |S_v| \le l$ where $l = \alpha_{\text{sep}}(k)$. Consider $\text{Ring}(S_u, S_v)$, and the linkage $\mathcal{P}(u, v)$ in $G$. Our goal is to modify $\mathcal{P}(u, v)$ to obtain another linkage $\mathcal{P}'(u, v)$ that is aligned with it and has a small winding number with respect to $\text{path}_R(u, v)$.
Recall that by Observation 7.4, we have an encircling tight sequence of concentric cycles $\mathcal{C}(u,v)$ in $\text{Ring}(S_u,S_v)$ that contains at least $40l$ cycles. Let $\eta(u,v)$ be the path in this ring witnessing the tightness of $\mathcal{C}(u,v)$. Further, recall the linkage $\text{Flow}R(u,v)$ between $S_u$ and $S_v$ in the subgraph $G{u,v}$ of $G$ (the restriction of $G$ to $\text{Ring}(S_u,S_v)$), and that $\text{Flow}R(u,v)$ is minimal with respect to $\mathcal{C}(u,v)$. We can assume w.l.o.g. that $\mathcal{P}(u,v)$ is minimal with respect to $\mathcal{C}(u,v)$. Otherwise, there is another solution $\hat{\mathcal{P}}$ such that it is identical to $\mathcal{P}$ in $G - V(S_u,S_v)$ and $\hat{\mathcal{P}}(u,v)$ is a minimal linkage with respect to $\mathcal{C}(u,v)$ (that is aligned with $\mathcal{P}(u,v)$). Then, we can consider $\hat{\mathcal{P}}$ instead of $\mathcal{P}$. Let $\mathcal{P}{\text{traverse}}(u,v)$ be the set of traversing paths in $\mathcal{P}(u,v)$. Since $\mathcal{P}{\text{traverse}}(u,v)$ is a flow between $S_u$ and $S_v$ in $G{u,v}$ and $\text{Flow}_R(u,v)$ is a maximum flow, clearly $|\text{Flow}R(u,v)| \ge |\mathcal{P}{\text{traverse}}(u,v)|$.