Subquadratic maximal-clique conjecture for locally chordal graphs

Let $G$ be a finite graph, and call it $r$-locally chordal if every ball of radius $r$ in $G$ is chordal. Let $|V(G)|$ denote the number of vertices of $G$ and let $O(|V(G)|^{1+\eps})$ have its usual asymptotic meaning. Maximal-clique conjecture. For every $\eps > 0$, there exists an integer $r \geqslant 0$ such that the number of maximal cliques in finite $r$-locally chordal graphs $G$ is $O(|V(G)|^{1+\eps})$.