Symmetry-group conjecture for graphs of type $3_n$

Let $3_n(G)$ denote a graph of type $3_n$ with point group $G$, where the groups considered are $D_{2h}$, $D_{2d}$, and $D_2$. A graph is tight when it satisfies the source's tightness condition.

Symmetry-group conjecture. (i) $3_n(D_{2h})$ exists if and only if $\frac{n}{4}$ is even and $n\geq 16$; there are no tight $3_n(D_{2h})$. (ii) $3_n(D_{2d})$ exists if and only if either $\frac{n}{4}$ is odd and $n\geq 20$, or $\frac{n}{8}$ is even and $n\geq 24$; the graph defined in Theorem Possible-forms-for-3nS (iii) is the unique tight $3_n(D_{2d})$ graph if $\frac{n}{4}$ is odd and $n\geq 20$. (iii) $3_n(D_2)$ exists if and only if $n\geq 24$ and $n\ne 28,32$; tight $3_n(D_2)$ exists if and only if $\frac{n}{4}$ is odd, $n\geq 44$, and $n\ne 60,84$; there are $i$ tight $3_n(D_2)$ starting with $\frac{n}{4}=6i+5$ for $1\leq i\leq 9$, except the case $i=5$ starting with $\frac{n}{4}=37$.

The authors say that they checked this conjecture for $n\leq 500$; its general validity is not resolved in the supplied text.