Estimates for the dilatation of $\sigma$-harmonic mappings

We consider planar $\sigma$-harmonic mappings, that is mappings $U$ whose components $u^1$ and $u^2$ solve a divergence structure elliptic equation ${\rm div} (\sigma \nabla u^i)=0$, for $i=1,2$. We investigate whether a locally invertible $\sigma$-harmonic mapping $U$ is also quasiconformal. Under mild regularity assumptions, only involving $\det \sigma$ and the antisymmetric part of $\sigma$, we prove quantitative bounds which imply quasiconformality.