The equality conjecture for Christoffel functions and birth-death processes

Let $C_n(\psi)$ be the Christoffel-function quantity associated with the normalized birth-death process, and let $\rho_n$ and $\eta$ be the quantities appearing in the asymptotic aperiodicity criterion. Assume that $C_n(\psi)$ has a limit as $n\to\infty$.

Equality conjecture. Then

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

This is stated as a stronger conjecture than the preceding implication. The existence of the right-hand limit is known, while the asserted equality remains open in the supplied text.