Eventual periodic representative conjecture for asymptotically periodic sequences

Let $S({\bf C}^{N})^{\infty}_{\mathrm{AP}}$ denote the asymptotically periodic sequences and let $S({\bf C}^{N})^{\infty}_{\mathrm{EP}}$ denote the eventually periodic sequences. Write $z\sim y$ for the equivalence relation on these sequences used in the paper. Eventual periodic representative conjecture. If $z$ is asymptotically periodic, then there is always $y\in S({\bf C}^{N})^{\infty}_{\mathrm{EP}}$ such that $z\sim y$. The conjecture addresses the unresolved decomposability question in the gray zone of noneventually periodic and asymptotically periodic sequences; the preceding lemma verifies the asserted equivalence for one explicit example, but the general claim is not established in the supplied text.