Connected Julia set dichotomy conjecture for rational functions

Let $f:\widehat{\mathbb C}\to\widehat{\mathbb C}$ be a rational function of degree at least $2$. Assume that the Julia set $J(f)$ is connected. Connected Julia set dichotomy conjecture. Then $f$ is either a finite Blaschke product in some holomorphic coordinates, a quotient of a Blaschke product by a rational function of degree $2$, or

$$\operatorname{HD}(J(f))>1.$$

A positive answer would generalize the stated theorem on Hausdorff dimension to rational maps. The supplied parser status is unknown, and the source context does not establish whether the conjecture has been resolved.