Circulant graph conjecture for monophonic position number two

Let $n$ be an integer, and let a circulant graph mean a graph whose vertices are arranged cyclically with adjacency determined by a fixed set of cyclic differences. The monophonic position number is denoted by $a$, and the diameter by $D$. Circulant graph conjecture. For any $n\geq 11$, there is a circulant graph with order $n$, monophonic position number $a=2$, and diameter $D=2$. The preceding theorem establishes existence of some graph with these parameters if and only if $n\in\{3,4,5,8\}$ or $n\geq 11$; the conjecture strengthens the large-order existence assertion by requiring the graph to be circulant, based on computational evidence.