Hausdorff-content comparability under principal quasiconformal mappings

Let $d\in(0,2]$, let $E\subset\mathbb C$ be compact, and let $h$ be a principal $K$-quasiconformal mapping conformal on $\mathbb C\setminus E$. Here $\mathcal M^d$ denotes $d$-dimensional Hausdorff content, and $A\simeq B$ means comparability up to multiplicative constants. Hausdorff-content comparability conjecture. One has

$$\mathcal M^d(h(E))\simeq\mathcal M^d(E),$$

with constants depending only on $K$ and $d$. Such a counterpart to the known one-dimensional estimate is presented as crucial for understanding distortion of Hausdorff measures under quasiconformal mappings, and remains open in the supplied text.