The equality of replication numbers for cycles and paths

Let $P_n$ and $C_n$ denote, respectively, the path and cycle on $n$ vertices, and let $\rho_R(G)$ denote the replication number of a graph $G$. The cycle–path replication conjecture. For every $n\geq 6$,

$$\rho_R(C_n)=\rho_R(P_n).$$

The conjecture is motivated by computations and an observed relationship between the replication numbers of cycles and paths; its validity for all $n\geq 6$ is left open in the source.