For all $w \in \text{Sep}_{R^2}(P, u)$, it holds that $\text{dist}_H(w, u'P) \le \alpha{\text{sep}}(k)$.
For all $w \in \text{Sep}_{R^2}(P, v)$, it holds that $\text{dist}_H(w, v'P) \le \alpha{\text{sep}}(k)$.
For all $w \in V(C_{R^2,P,u})$, it holds that $\text{dist}H(w, \tilde{A}{R^2,P,u} \cup {u'P}) \le \alpha{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
For all $w \in V(C_{R^2,P,v})$, it holds that $\text{dist}H(w, \tilde{A}{R^2,P,v} \cup {v'P}) \le \alpha{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
For all $w \in V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}) \cup \text{Sep}{R^2}(P, u) \cup \text{Sep}{R^2}(P, v))$, it holds that $\text{dist}H(w, V(R^2) \setminus (\tilde{A}{R^2,P,u} \cup \tilde{A}{R^2,P,v})) \le \alpha{\text{dist}}(k) + \alpha_{\text{sep}}(k)$.
Proof. First, note that Conditions 1 and 2 follow directly from Lemma 6.4 and Observation 6.6.
For Condition 3, consider some vertex $w \in V(C_{R^2,P,u})$. By Lemma 6.1, $\text{dist}H(w, V(R^2)) \le \alpha{\text{dist}}(k)$. Thus, there exists a path $Q$ in $H$ with $w$ as one endpoint and the other endpoint $x$ in $V(R^2)$ such that the length of $Q$ is at most $\alpha_{\text{dist}}(k)$. In case $x \in \tilde{A}{R^2,P,u} \cup {u'P}$, we have that $\text{dist}H(w, \tilde{A}{R^2,P,u}) \le \alpha{\text{dist}}(k)$, and hence the condition holds. Otherwise, by the definition of $\text{Sep}{R^2}(P, u)$, the path $Q$ must traverse at least one vertex from $\text{Sep}{R^2}(P, u)$. Thus, $\text{dist}H(w, \text{Sep}{R^2}(P, u)) \le \alpha{\text{dist}}(k)$. Combined with Condition 1, we derive that $\text{dist}H(w, \tilde{A}{R^2,P,u} \cup {u'P}) \le \alpha{\text{dist}}(k) + \alpha_{\text{sep}}(k)$. The proof of Condition 4 is symmetric.
The proof of Condition 5 is similar. Consider some vertex $w \in V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}) \cup \text{Sep}{R^2}(P, u) \cup \text{Sep}{R^2}(P, v))$. As before, there exists a path $Q$ in $H$ with $w$ as one endpoint and the other endpoint $x$ in $V(R^2)$ such that the length of $Q$ is at most $\alpha_{\text{dist}}(k)$. In case $x \in \tilde{A}{R^2,P,u} \cup {u'P}$, we are done. Otherwise, the path $Q$ must traverse at least one vertex from $\text{Sep}{R^2}(P, u) \cup \text{Sep}{R^2}(P, v)$. Specifically, if $x \in V(C_{R^2,P,u})$, then it must traverse at least one vertex from $\text{Sep}{R^2}(P, u)$, and otherwise $x \in V(C{R^2,P,v})$ and it must traverse at least one vertex from $\text{Sep}{R^2}(P, v)$. Combined with Conditions 1 and 2, we derive that $\text{dist}H(w, V(R^2) \setminus (\tilde{A}{R^2,P,u} \cup \tilde{A}{R^2,P,v})) \le \alpha_{\text{dist}}(k) + \alpha_{\text{sep}}(k)$. $\square$
An immediate corollary of Lemma 6.9 concerns the connectivity of the “middle region” as follows. (This corollary can also be easily proved directly.)
Corollary 6.2. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour, $P$ be a long maximal degree-2 path of $R^2$, and $u$ and $v$ be its endpoints. Then, $H[V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))]$ is a connected graph.
Proof. By Lemma 6.9 and the definition of $\text{Sep}{R^2}(P, u)$ and $\text{Sep}{R^2}(P, v)$, for every vertex in $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))$, the graph $H$ has a path from that vertex to some vertex in $\text{Sep}{R^2}(P, u) \cup \text{Sep}{R^2}(P, v)$ that lies entirely in $H[V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v}))]$. Thus, the corollary follows from Observation 6.6. $\square$
Next, we utilize Lemma 6.9 and Corollary 6.2 to argue that the “middle regions” of different long maximal degree-2 paths of $R^2$ are distinct. Recall that we wish to reroute a given solution to be a solution that “spirals” only a few times around the Steiner tree. This lemma allows us to independently reroute the solution in each of these “middle regions”. In fact, we prove the following stronger statement concerning these regions. The idea behind the proof of this lemma is that if it were false, then $R^2$ admits a detour, which is contradiction.
Lemma 6.10. Let $(G, S, T, g, k)$ be a good instance of Planar Disjoint Paths. Let $R^2$ be a Steiner tree that has no detour. Additionally, let $P$ and $\hat{P}$ be two distinct long maximal degree-2 paths of $R^2$. Let $u$ and $v$ be the endpoints of $P$, and $\hat{u}$ and $\hat{v}$ be the endpoints of $\hat{P}$. Then, one of the two following conditions holds:
• $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{u}}).$
• $V(H) \setminus (V(C_{R^2,P,u}) \cup V(C_{R^2,P,v})) \subseteq V(C_{R^2,\hat{P},\hat{v}}).$