that will be needed. For the remainder of this paper, we let $X$ be a metric surface and $Q$ denote a fixed quadrilateral in $X$. We write $\Gamma_1$ for $\Gamma_1(Q)$. We assume throughout this paper that all curves are non-constant.

For $k \in \{1,2\}$ and $\varepsilon > 0$, the [*$k$-dimensional Hausdorff $\varepsilon$-content*]{} of a set $E \subset X$, denoted by $\mathcal{H}_\varepsilon^k(E)$, is defined as $$\mathcal{H}_\varepsilon^k(E) = \inf \left\{ \sum a_k \operatorname{diam}(A_j)^k: E \subset \bigcup_{j=1}^\infty A_j, \operatorname{diam}A_j < \varepsilon \right\},$$ with normalizing constants $a_1 = 1$ and $a_2 = \pi/4$. The [*Hausdorff $k$-measure*]{} of $E$ is defined as $\mathcal{H}^k(E) = \lim_{\varepsilon \rightarrow 0} \mathcal{H}_\varepsilon^k(E)$.

We proceed with an overview of the construction of the harmonic function $u$ corresponding to the curve family $\Gamma_1$, as given in Section 4 of [@Raj:16]. By a standard argument using Mazur’s lemma, there exists a sequence of admissible functions $(\rho_k)$ for $\Gamma_1$ that converges strongly in $L^2$ to a function $\rho \in L^2(Q)$ satisfying $\int_Q \rho^2\,d\mathcal{H}^2 = \operatorname{mod}\Gamma_1$. By Fuglede’s lemma, $$\label{equ:fuglede}
    \int_\gamma \rho_k\, ds \rightarrow \int_{\gamma} \rho\, ds < \infty$$ for all curves $\gamma$ in $Q$ except for a family of modulus zero. In particular, this implies that $\rho$ is weakly admissible for $\Gamma_1$ (that 