The harmonic-measure support conjecture for graphs with the doubling property

Let $G$ be an infinite graph with the doubling property: there exists a universal constant $C>0$ such that for every $r>0$ and every vertex $x$,

$$|B(x,2r)|\leq C|B(x,r)|,$$

where $B(x,r)$ is the ball of radius $r$ around $x$ in the graph metric. For a subset $S\subset B(x,r)$ and a vertex $z\notin B(x,r)$, consider the harmonic measure of $S$ from $z$. Harmonic-measure support conjecture. As $r\to\infty$, $1-o(1)$ of this harmonic measure is supported on a subset of $S$ of size $o(|B(x,r)|)$. This predicts strong concentration of harmonic measure in graphs with volume doubling, extending the expected small-support behavior from polynomial-growth settings; the supplied text does not indicate whether the claim has been proved or disproved.