Prime-distance characterization of circulant graphs

For integers $n$ and $k$ with $k\leq n/2$, let $\operatorname{Circ}(n,k)$ denote the circulant graph with these parameters. A graph is a prime distance graph if its vertices can be labeled by distinct integers so that adjacent vertices receive labels whose absolute difference is prime. Prime-distance circulant conjecture. The circulant graph $\operatorname{Circ}(n,k)$ is a prime distance graph if and only if none of the following hold:

1. $n$ is odd and $k=2$.
2. $n$ is odd and $k=(n-1)/2$.
3. $(n,k)=(6,2)$.

The claim extends the preceding classification theorem to the remaining parameter values; it was verified in the source for $n\leq 14$, but is otherwise open.