Thomas's cage monophonic position conjecture

Let a $(d,g)$-cage $G$ be a graph of degree $d$ and girth $g$, and let $\operatorname{mp}(G)$ denote its monophonic position number. Thomas's conjecture. For sufficiently large $g$, the monophonic position number of a $(d,g)$-cage $G$ satisfies

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

The survey presents this as a computationally suggested conjecture, and no resolution is given.