Near-linear bound on maximal cliques in locally chordal graphs

Prove that for every ε > 0, there exists an integer r > 0 such that every finite r-locally chordal graph G has O(|V(G)|^{1+ε}) maximal cliques.

Background

Finite chordal graphs on n vertices have at most n maximal cliques. The paper observes that every finite r-locally chordal graph with r ≥ 3 has at most n² maximal cliques, because each maximal clique is contained in a 3/2-ball and every such ball is chordal.

The authors conjecture that the dependence on r can substantially improve this quadratic bound: for any fixed ε > 0, sufficiently large locality parameter r should force the number of maximal cliques to be O(n^{1+ε}). They note that the conjectured behavior holds for graphs of girth greater than r, where the number of maximal cliques equals the number of edges and known extremal bounds give a subquadratic estimate.