Helly–Erdős–Pósa property for locally chordal region representations

Determine whether, for every integer r ≥ 3 and every r-locally chordal graph G, the family of regions in every r-acyclic region representation of G satisfies the Helly–Erdős–Pósa property.

Background

For chordal graphs, the subtrees arising from acyclic region representations satisfy the stronger Helly–Erdős–Pósa property: every finite subfamily either contains k pairwise disjoint members or can be intersected by a vertex set of size at most k−1.

The paper proves only the ordinary Helly property for the families associated with r-acyclic region representations of r-locally chordal graphs. It asks whether the stronger packing-versus-transversal conclusion also holds in this locally chordal setting.