The conjecture on monophonic position numbers of cages

A $(d,g)$-cage is a graph of minimum possible order among graphs with degree $d$ and girth $g$, and ${\rm mp}(G)$ denotes the monophonic position number of $G$. The cage monophonic position conjecture. For sufficiently large $g$, the monophonic position number of a $(d,g)$-cage $G$ satisfies

$${\rm mp}(G)<d.$$

This conjecture is motivated by the computed monophonic position numbers of the Petersen, Heawood, and McGee graphs, which are three, three, and two, respectively. It predicts that for cages of sufficiently large girth, the monophonic position number is strictly smaller than the degree.