Invariant graph conjecture for marked rational maps

Let $(f,P)$ be a marked rational map, with Julia set $J_f$, and let $G$ be a graph in the sphere. An $f$-invariant graph satisfies $f(G)=G$; the post-critical set is denoted by $P$. Invariant graph conjecture. Theorem should hold with $n=1$: there exists an $f$-invariant graph

$$G\subset J_f$$

such that

$$P\cap J_f\subset G$$

and each component of

$$\overline{\mathbb C}\setminus G$$

contains at most one point of $P$. The conjecture would follow in particular from the corresponding invariant-graph statement for marked expanding Thurston maps; the paper explains that it is known under certain Julia-set hypotheses, but leaves the general case open.