case. We may assume without loss of generality that $E$ does not intersect $\zeta_4$. Proposition \[prop:coarea\] applied to $g(x)=\operatorname{dist}(\zeta_4,x)$ shows that there exists a small $p>0$ such that $\mathcal{H}^1(g^{-1}(p))< \infty$. Moreover, by Lemma \[lemm:separating\_continuum\] there is a continuum $F \subset g^{-1}(p)$ joining $\zeta_1$ and $\zeta_3$ in $Q \setminus E$. Proposition \[prop:existence\_paths\] gives a simple curve $\gamma$ such that $|\gamma| \subset F$ also joins $\zeta_1$ and $\zeta_3$. As before, we may assume that $\gamma \notin \Gamma_0$ so that $u||\gamma|$ takes all values between $0$ and $1$. This is a contradiction since $|\gamma| \cap E = \emptyset$. The proof is complete.

Next, we give a generalization of Lemma 15.6 in [@Raj:16], with a corrected constant. The proof is essentially the same as the corresponding proof in [@Raj:16].

\[lemm:oscillation\] Let $x \in Q$ and $r \in (0, r_0)$, where $r_0 = \min\{\operatorname{diam}\zeta_1, \operatorname{diam}\zeta_3\}/4$. Then $$\label{equ:oscillation_bound}
        r  \mathcal{H}^1(u(B(x,r) \cap Q))\leq \frac{4}{\pi} \int_{B(x,2r)} \rho\, d\mathcal{H}^2.$$ Moreover, if $U(x,r)$ is the $x$-component of $B(x,r) \cap Q$, then $$\label{equ:oscillation_bound2}
        r  \operatorname*{osc}_{U(x,r)} u \leq \frac{4}{\pi} \int_{B(x,2r)} \rho\, d\mathcal{H}^2.$$

By applying Proposition \[prop:coarea\] to the function $d(\cdot,x)$ and arguing as in the first paragraph of the proof of Proposition \[prop:connect\], we see that for almost every $s \in (r,2r)$, 