The cycle-to-path local metamorphosis conjecture

Let $k$ be a positive integer, and consider parallel classes whose blocks are copies of the cycle $C_k$ or the path $P_k$. Cycle-to-path local metamorphosis conjecture. The union of $k-1$ parallel classes of $C_k$ is decomposable into $k$ parallel classes of $P_k$. This conjecture would generalize the local metamorphoses established in Theorems 1 and 4, but the supplied text gives no resolution.