is reciprocal (and hence $X$ admits a quasiconformal parametrization), then Theorem \[thm:main\] holds with $\kappa = (4/\pi)^2$. For this reason, it is natural to conjecture that the best possible $\kappa$ for the general case is also $(4/\pi)^2$, though our techniques fall far short of this.

In Proposition 15.8 of [@Raj:16], Theorem \[thm:main\] (with a larger value of $\kappa$) is proved under the assumption that $X$ satisfies the mass upper bound $\mathcal{H}^2(B(x,r)) \leq Cr^2$ for some $C>0$ independent of $x$ and $r$. Our proof follows a similar outline; the difficulty is to avoid using the upper bound.

The basic approach is to construct an “energy-minimizing” or “harmonic” function $u: Q \rightarrow [0,\infty)$ which satisfies the boundary constraints $u|\zeta_1 = 0$ and $u|\zeta_3 = 1$. Working only from the assumptions at hand, one can establish relevant properties of $u$. The main property needed to prove Theorem \[thm:main\] is that a version of the coarea inequality holds for $u$. For the case when $X$ satisfies the mass upper bound $\mathcal{H}^2(B(x,r)) \leq Cr^2$, this is found in Proposition 15.7 of [@Raj:16]. The coarea inequality implies that, from the level sets of $u$, one may extract a large family of rectifiable curves contained in $\Gamma_2(Q)$. 