Quasi-stationary extinction-event conjecture

Let $\mu_N$ be the quasi-stationary distribution of the Markov chain, let $\mathcal E_N$ be the extinction event defined in Theorem~, and let $P_{\mu_N}$ denote probability when the initial state has distribution $\mu_N$. Assume the conditions of Theorem~.

Quasi-stationary extinction-event conjecture.

$$\lim_{N\to\infty}P_{\mu_N}(\mathcal E_N)=1.$$

The paper states that this is almost equivalent to the preceding conjecture's asymptotic formula for $1-\lambda_N$, via the probability that the chain exits through the set of vertices indexed by the fittest types. It is presented as an equivalent reformulation and remains unproved.