Polynomial-time computability of Julia sets without critical points in the postcritical set

Let $f$ be a rational map. Write $J_f$ for its Julia set and $\Omega_f$ for its postcritical set. A Turing machine has an oracle for the coefficients of $f$ if it can access those coefficients through oracle queries.

Computability conjecture. If $\Omega_f$ does not contain any critical points, then $J_f$ is polynomial-time computable by a Turing machine with an oracle for the coefficients of $f$.

The algorithm proving the paper's main theorem does not apply when $\Omega_f$ contains a parabolic periodic point, but the authors expect this broader computability statement to hold.