also recall Lemma 4.5 of [@Raj:16], where it is shown that $0 \leq u(x) \leq 1$ for all $x \in Q$. It follows from that if $x \in \zeta_3$ lies in the image of a curve $\gamma \in \Gamma_1 \setminus \Gamma_0$, then $u(x) \geq 1$ and thus $u(x) = 1$.

As final points of notation, for a set $A \subset Q$, let $\operatorname*{osc}_{A} u = \sup_{x,y \in A} |u(x) - u(y)|$. Let $|\gamma|$ denote the image of the curve $\gamma$ in $Q$.

To study the harmonic function $u$, there are three auxiliary results which are employed repeatedly in [@Raj:16] and which we state here for easy reference. The first concerns the existence of rectifiable curves and can be found as Proposition 15.1 of [@Sem:96c].

\[prop:existence\_paths\] Let $x,y \in X$ be given, $x \neq y$. Suppose that $E \subset X$ is a continuum with $\mathcal{H}^1(E) < \infty$ and $x, y \in E$. Then there is an $L>0$, $L \leq \mathcal{H}^1(E)$, and an injective 1-Lipschitz mapping $\gamma\colon [0,L] \rightarrow X$ such that $\gamma(t) \in E$ for all $t$, $\gamma(0) = x$, $\gamma(L) = y$ and $\mathcal{H}^1(\gamma(F)) = \mathcal{H}^1(F)$ for all measurable sets $F \subset [0,L]$.

The next is the standard coarea inequality 