Monophonic position in cages

Prove or disprove that, for sufficiently large girth \(g\), every \((d,g)\)-cage \(G\) satisfies \(\mono(G)<d\).

Background

A (d,g)(d,g)-cage is a smallest graph of girth gg and minimum degree dd. The survey records a conjecture that sufficiently large girth forces the monophonic position number below the degree parameter.

References

For sufficiently large $g$, the monophonic position number of a $(d,g)$-cage $G$ satisfies $\mono (G) < d$.

— The General Position Problem: A Survey  (2501.19385 - V. et al., 31 Jan 2025) in Section 5, subsection “Monophonic position sets”