Astala's conjecture on the sharp dimension of quasicircles

Let $0<k<1$, and let a $k$-quasicircle be a quasicircle with parameter $k$. Astala's conjecture. There exists a $k$-quasicircle $\Gamma$ with Hausdorff dimension

$$\dim \Gamma=1+k^2.$$

Astala's upper estimate $\dim\Gamma\leq 1+k^2$ is known, while the conjecture asks for sharpness, namely attainment of the bound for every $0<k<1$.