Even-diameter order conjecture for chordal multi-ring mixed graphs

Let $CMRM(m,n,c)$ be a chordal $m$-ring mixed graph on vertex set $\mathbb Z_m\times\mathbb Z_n$, with the arcs and chords defined by the source, and let $k$ be an even diameter satisfying $k\equiv 2\pmod 4$. Chordal multi-ring mixed-graph conjecture. The maximum number $N$ of vertices is

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

and this value is attained with $m=2$ cycles and chord

$$c=\frac{k}{2}.$$

The corresponding single-ring mixed-graph conjecture gives the smaller value $k(\frac{k}{2}-1)+4$ in this congruence class; the multi-ring construction is proposed to attain the larger value.