Quasiconformal parametrization of metric surfaces with small dilatation

We verify a conjecture of Rajala: if $(X,d)$ is a metric surface of locally finite Hausdorff 2-measure admitting some (geometrically) quasiconformal parametrization by a simply connected domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$, then there exists a quasiconformal mapping $f:X\to \mathrm{\Omega }$ satisfying the modulus inequality $2\pi^{-1}$\text{Mod }$\Gamma \leq $\text{Mod }$f\Gamma \leq 4\pi^{-1}$\text{Mod }$\Gamma$forallcurvefamilies$\Gamma$in$X$. This inequality is the best possible. Our proof is based on an inequality for the area of a planar convex body under a linear transformation which attains its Banach-Mazur distance to the Euclidean unit ball.