Polynomial-time computability of Julia sets under the critical-orbit dichotomy

Let $f$ be a rational map of degree $d\geqslant 2$, let $J_f$ denote its Julia set, and let $Df^n$ denote the derivative of the $n$-th iterate. Assume the following two conditions:

1. the forward orbit of every critical point $c\notin J_f$ either is finite or converges to an attracting periodic orbit;
2. for any critical point $c\in J_f$, either there exist $\gamma,C>0$ such that

$$|Df^n(f(c))|\geqslant Ce^{\gamma n}\quad\text{for any }n\in\mathbb{N},$$

or the forward orbit of $c$ contains another critical point.

Computability conjecture. Under these two conditions, $J_f$ is polynomial-time computable by a Turing machine with an oracle for the coefficients of $f$.

This is presented as a generalization of the paper's main computability theorem. Together with the preceding typical critical-orbit conjecture, it motivates the expectation that Julia sets are polynomial-time computable for almost all rational maps, but the source gives no proof of this generalization.