Logarithmic quasi-stationary extinction-rate conjecture

Let $\lambda_N$ be the quasi-stationary eigenvalue of the interior transition matrix, and let $\alpha$ be the parameter from Theorem~. Assume the conditions of Theorem~.

Logarithmic extinction-rate conjecture.

$$\lim_{N\to\infty}\frac{1}{N}\log(1-\lambda_N)=\alpha.$$

This is presented as a weak form of the conjectured sharp asymptotic relation $\alpha^N/(1-\lambda_N)\to1$. It gives only the exponential scale of the quasi-stationary extinction probability and is not proved in the paper.