Computable-modulus conjecture for the conformal-radius function

Let $f$ be the $1$-periodic continuous function

$$f(\theta)=\Phi(\theta)+\log r(\theta).$$

A computable modulus of continuity for $f$ is a computable function $\mu:\mathbb{N}\to\mathbb{N}$ such that

$$|f(\theta_1)-f(\theta_2)|<2^{-n}\quad\text{whenever}\quad |\theta_1-\theta_2|<2^{-\mu(n)}.$$

Computable-modulus conjecture. The function $f$ has a computable modulus of continuity.

This conjecture would provide the quantitative continuity needed to show that the explicitly constructed parameter $\theta$, whose Julia set is non-computable, is nevertheless computable in polynomial time. No restriction is imposed on the growth rate of $\mu$ beyond computability; the source gives no resolution of the conjecture.