and establish some notation. Let $(X,d,\mu)$ be a metric measure space. For a family $\Gamma$ of curves in $X$, the [*$p$-modulus*]{} of $\Gamma$ is defined as $$\operatorname{mod}_p \Gamma = \inf \int_X \rho^p\,d\mu ,$$ where the infimum is taken over all Borel functions $\rho: X \rightarrow [0,\infty]$ with the property that $\int_{\gamma} \rho\,ds \geq 1$ for all locally rectifiable curves $\gamma \in \Gamma$. Such a function $\rho$ is called [*admissible*]{}. If the exponent $p$ is understood, a homeomorphism $f: (X,d,\mu) \rightarrow (Y,d',\nu)$ between metric measure spaces is [*quasiconformal*]{} if there exists $K \geq 1$ such that $$K^{-1} \operatorname{mod}_p \Gamma \leq \operatorname{mod}_p f(\Gamma) \leq K \operatorname{mod}_p \Gamma$$ for all curve families $\Gamma$ in $X$. In this paper, we always take $p=2$ and assume that a metric space $(X,d)$ is equipped with the Hausdorff 2-measure $\mathcal{H}^2$, and we write $\operatorname{mod}\Gamma$ in place of $\operatorname{mod}_2 \Gamma$.

Throughout this paper, we assume that $(X,d)$ is a metric surface as defined above. A [*quadrilateral*]{} in $X$ is a subset $Q \subset X$ homeomorphic to $[0,1]^2$ with four designated non-overlapping boundary arcs, denoted in cyclic order by $\zeta_1$, $\zeta_2$, $\zeta_3$, $\zeta_4$, which are the images of $[0,1] \times\{0\}$, $\{1\} \times [0,1]$, $[0,1] \times \{1\}$ and $\{0\} \times 