The NILF conjecture for Julia sets of rational maps

Let $f$ be a rational map, and let $J(f)$ denote its Julia set. An invariant line field on $J(f)$ is an invariant measurable field of unoriented lines defined on $J(f)$. NILF conjecture. The Julia set $J(f)$ does not support an invariant line field unless $f$ is a Lattès example.

This conjecture concerns the rigidity of rational maps: the absence of invariant line fields is related to the equivalence between topological or quasiconformal conjugacy and conformal conjugacy. Its status is not resolved by the supplied context.