be shown to be quasiconformal. However, the question of whether a weaker set of assumptions might still be sufficient to construct such a quasiconformal mapping is not fully settled in [@Raj:16].

It is not difficult to construct examples of metric surfaces for which conditions and fail. For instance, the quotient space $\mathbb{R}^2/ \sim$, where $x \sim y$ if $x=y$ or if both $x$ and $y$ belong to the closed unit disc, has a natural metric for which both conditions fail. On the other hand, it was conjectured in [@Raj:16] (Question 17.5) that in fact condition holds for all $(X,d)$. The main result of this paper shows that this is indeed the case.

\[thm:main\] Let $(X,d)$ be a metric space homeomorphic to $\mathbb{R}^2$ with locally finite Hausdorff 2-measure. There exists a constant $\kappa\geq 1$, independent of $X$, such that $\operatorname{mod}\Gamma_1(Q) \operatorname{mod}\Gamma_2(Q) \geq 1/\kappa$ for all quadrilaterals $Q \subset X$.

As a consequence of Theorem \[thm:main\], condition in Definition \[defi:reciprocality\] is unnecessary. Our proof as written gives a value of $\kappa = 2000^2\cdot (4/\pi)^2$, though optimizing each step would improve this to $\kappa = 216^2\cdot (4/\pi)^2$. It is a corollary of Theorem 1.5 in [@Raj:16], as improved in [@Rom:17], that if $X$ 