The cycle independence-equivalence conjecture

For a positive integer $n$, let $C_n$ be the cycle graph on $n$ vertices, let $D_n$ be the graph obtained from $C_n$ by adding a loop structure as defined in the paper, and let $[C_n]$ denote the independence equivalence class of $C_n$, namely the graphs having the same independence polynomial as $C_n$. Cycle independence-equivalence conjecture. If $3\!\!\!\not\vert n$ and $n\ge 4$, then

$$[C_n]=\{C_n,D_n\}.$$

This conjecture gives the expected complete description of the independence equivalence classes of cycles whose lengths are not divisible by $3$. The paper notes that multiples of $3$ can have triangle components and that the behavior of such cycles is more complicated; the proposed statement concerns the remaining cycle lengths.