Makienko's conjecture on buried points in rational Julia sets

Let $R:{\mathbb C_\infty}\rightarrow {\mathbb C_\infty}$ be a rational function. A point of the Julia set $J(R)$ is buried if it does not belong to the boundary of a Fatou component. A Fatou component of $R^2$ is completely invariant if its image and full preimage under $R^2$ are itself.

Makienko's conjecture. The Julia set $J(R)$ has buried points if and only if there is no completely invariant component of the Fatou set of $R^2$.

Known examples with non-empty residual Julia sets have no Fatou component with a finite grand orbit, and the conjecture proposes an exact dynamical criterion for the existence of buried points. Its resolution is not specified in the source.