Invariant graph conjecture for marked expanding Thurston maps

Let $(F,Q)$ be a marked expanding Thurston map, where $F$ is an expanding Thurston map of the sphere and $Q$ is the marked set. An $F$-invariant graph is a graph preserved by $F$. Invariant graph conjecture. There exists an $F$-invariant graph containing $Q$. This conjecture, attributed in the source to a problem of Bonk and Meyer, would imply the rational-map conjecture above in the Sierpiński-carpet and sphere Julia-set cases; its general status is not resolved in the supplied text.