The no invariant line field conjecture for rational maps

Let $f$ be a rational map, let $J$ be its Julia set, and let an invariant line field on $J$ mean a line field fixed by the pushforward operator

$$f^*: \nu\mapsto (\nu\circ f)\frac{|f'|^2}{(f')^2}.$$

A rational map is double covered by an integral torus endomorphism in the exceptional case described in the source. No invariant line field conjecture. A rational map $f$ carries no invariant line field on its Julia set $J$, except when $f$ is double covered by an integral torus endomorphism. In particular, the conjecture implies that there is no invariant line field on $J$ when the Fatou set $F:=\widehat{\mathbb C}\setminus J$ is non-empty, for example when $f$ is a polynomial. This conjecture is stronger than Fatou's density of hyperbolicity conjecture and shifts attention from parameter families to the dynamics of individual maps; the supplied text does not state a resolution.