Beaton–Brown–Cameron conjecture on independence equivalence of odd cycles

Let $n\geq 4$ be odd, let $C_n$ be the cyclic graph on $n$ vertices, and let $D_n$ be the graph shown in Figure~. Two graphs are independence equivalent when they have the same independence polynomial.

Beaton–Brown–Cameron conjecture. If $3\not\vert n$, then a graph $G$ is independence equivalent to $C_n$ if and only if

$$G\in\{C_n,D_n\}.$$

This conjecture asks for a complete description of the independence equivalence class of $C_n$ for odd $n\geq 4$ not divisible by $3$. The cited results establish the analogous classification for even $n\geq 4$, except $n=6$, and for prime powers $n=p^k$ with $p\geq 5$ prime; the stated odd-cycle case remains open here.