Dimension drop for harmonic measure of conformal self-similar sets

Let $J$ be a conformal self-similar set of dimension $\delta<1$, and let $\omega_{\mathbb C\setminus J}$ denote harmonic measure for its complement. Dimension-drop conjecture. The harmonic measure has strictly smaller dimension:

$$\dim\omega_{\mathbb C\setminus J}<\delta.$$

The source describes this as an extension of Carleson's result and says that it would follow from the preceding Green-function conjecture, but it remains unresolved in the stated generality.