Mandatory-vertex characterization for chordal graphs

Determine whether every chordal graph G satisfies |Man(G)|=meg(G), where Man(G) is the set of mandatory vertices and meg(G) is the monitoring edge-geodetic number.

Background

The paper establishes that |Man(G)|=meg(G) for strongly chordal graphs and notes that the same property was previously established for well-partitioned chordal graphs. Since strongly chordal and well-partitioned chordal graphs are subclasses of chordal graphs, the paper leaves unresolved whether the equality extends to all chordal graphs.

References

Does this property hold for chordal graphs as well?

— Characterizing optimal monitoring edge-geodetic sets for some structured graph classes  (2503.06086 - Foucaud et al., 8 Mar 2025) in Question 2, Section 6, “Conclusion and future aspects”