The invariant measurable line-field formulation of the density conjecture

Let ${\cal F}$ be the holomorphic family, let $B$ be a disk in the set of quasiconformally stable parameters, and let $J_\lambda$ denote the Julia set of the corresponding map. An equivalent density conjecture. There is no disk $B$ such that

$$J_\lambda=\widehat{\mathbb C}$$

for every $\lambda\in B$ and $J_\lambda$ carries an invariant measurable line field. This formulation expresses the obstruction to density through invariant measurable line fields on full-sphere Julia sets; the supplied text states that it is equivalent to the density conjecture, but gives no resolution.