Even-diameter order conjecture for chordal ring mixed graphs

Let $CRM(N,c)$ be a chordal ring mixed graph with vertex set $\mathbb Z_N$, directed-cycle arcs $i\mathbin{\rightarrow}i+1$, and chords determined by the odd parameter $c$ as in the source. Let $k>2$ be an even diameter. Even-diameter chordal-ring mixed-graph conjecture. The maximum number $N$ of vertices is

$$N=\frac{1}{2}k^2+2$$

if $k\equiv 0\pmod 4$, with

$$c=\frac{1}{4}n^2-\frac{1}{2}n+1,$$

and is

$$N=k\left(\frac{k}{2}-1\right)+4$$

if $k\equiv 2\pmod 4$. The general upper bound is attained for odd diameters but not for even $k>2$; this conjecture proposes the optimal even-diameter values based on prior work and computer exploration.