Mandatory-vertex characterization for permutation graphs

Determine whether every permutation 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 proves that for every bipartite permutation graph G, the mandatory-vertex set satisfies Man(G)=V\setminus Cut(G), and therefore |Man(G)|=meg(G). It also notes that this stronger equality with the non-cut vertices does not extend to all permutation graphs, citing a permutation graph formed from a six-vertex path by adding a universal vertex. The unresolved question is whether the weaker equality |Man(G)|=meg(G) nevertheless holds throughout the class of permutation graphs.

References

However, do we have $|Man(G)|=meg(G)$ when $G$ is a permutation graph?

— Characterizing optimal monitoring edge-geodetic sets for some structured graph classes  (2503.06086 - Foucaud et al., 8 Mar 2025) in Question 1, Section 6, “Conclusion and future aspects”