Super-base graph conjecture for the graphs $G_{n,k}$

Let $n\ge 5$ with $n\equiv 0,5\pmod 6$ and $k=\lfloor (n+2)/6\rfloor$. Let $G_{n,k}$ be the graph from the paper, and let $\overline{G_{n,k}}\setminus v$ denote the graph obtained by deleting any vertex $v$ from its complement. A minimal prime graph is a graph with the minimal-prime property defined in the paper.

Super-base graph conjecture. For every such $n$ and $k$, if $G_{n,k}$ is a minimal prime graph, then $\overline{G_{n,k}}\setminus v$ is not a minimal prime graph for any vertex $v$; consequently, these graphs cannot be generated from any minimal prime graph.

Computations for $n\le 100$ support the claim. It asserts that the indicated minimal prime graphs are super base graphs, meaning that they are not generated from any minimal prime graph.