Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ of $h$, is a diffeomorphism of $D$ and homeomorphism of $\bar{D}$.

My question is : suppose $h:S^1\to S^1$ is a k-quasisymmetric homeomorphism, then is the complex harmonic extension $H=E(h)\hspace{2mm} K$-quasiconformal with $K$ depending only on $k$ ?

Answers/resources/papers related to this would be greatly appreciated !