The normalized SRLP conjecture for birth-death processes

Let $\mathcal{X}$ be a birth-death process and let $\mathcal{\tilde{X}}$ be its normalized version. Let $C_n(\psi)$ be the associated quantity and let $\rho_n$ and $\eta$ be as in the preceding definitions. The normalized process has the SRLP when $\lim_{n\to\infty}C_n(\psi)=0$, and asymptotic aperiodicity is characterized by

$$\lim_{n\to\infty}\frac{\rho_n(-\eta)}{\rho_n(\eta)}=0.$$

Normalized SRLP conjecture. We have

$$\lim_{n\to\infty}C_n(\psi)=0\quad\Rightarrow\quad\lim_{n\to\infty}\frac{\rho_n(-\eta)}{\rho_n(\eta)}=0.$$

The right-hand limit exists, and the converse implication is already known; this conjecture is equivalent to an earlier conjecture attributed in the source to [DS95b].