Even-diameter order conjecture for chordal ring graphs

Let $CR(N,c)$ be a chordal ring graph with diameter $k$, and let $N(k)$ denote the maximum number of vertices among chordal ring graphs of diameter $k$. For even $k>2$, the conjecture concerns the largest possible order of such a graph. Even-diameter chordal-ring conjecture. The maximum number $N$ of vertices of a chordal ring graph with an even diameter $k>2$ is

$$N=\frac{3k^2}{2}-k.$$

The bound $N(k)=\frac{3k^2}{2}$ is known for even diameter, but cannot be attained when $k>2$; the conjecture proposes the exact next-best value.