Thomas's cubic-graph monophonic position conjecture

Let $G$ be a cubic graph of order $n$, and let $\operatorname{mp}(G)$ denote its monophonic position number. Thomas's conjecture. The largest possible monophonic position number of a cubic graph with order $n$ is

$$\frac{n}{3}+O(n).$$

This conjecture was suggested by computational results; the stated asymptotic form remains unresolved in the survey.