Dimension drop conjecture for harmonic measure on Fuchsian limit sets

Let $G$ be the group of isometries of the hyperbolic plane $\mathbb{H}^2$. Let $\mu$ be a finitely supported probability on $G$ whose support generates, as a semigroup, a discrete non-elementary subgroup $\Gamma$, and let $\Lambda$ be the limit set of $\Gamma$. Let $\nu$ be the unique $\mu$-stationary probability measure on $\Lambda$, and write $\dim(\nu)$ and $\dim(\Lambda)$ for its exact dimension and the Hausdorff dimension of $\Lambda$, respectively. Dimension drop conjecture. One has

$$\dim(\nu)<\dim(\Lambda).$$

The conjecture asserts that equality between the dimension of harmonic measure and the Hausdorff dimension of the limit set never occurs for finitely supported random walks; it is motivated by dimension-drop results for harmonic measure in other settings and remains open here.