Almost-sure persistence under mutual invasibility

Consider the two-species Markov chain

satisfying conditions B1--B2. Assume the single-species boundary dynamics satisfy the stated nontriviality condition, and each $F_i(x,\omega)$ is non-decreasing in both species abundances and positive whenever $x_i>0$. Let $\lambda_i(\mu_j)$ denote the mean per-capita growth rate of species $i$ at the invariant measure $\mu_j$ of the other species. **Chesson--Ellner persistence conjecture.** If $\lambda_1(\mu_2)>0$ and $\lambda_2(\mu_1)>0$, then

is almost surely persistent. Mutual invasibility is known to imply persistence in probability under the monotonicity assumptions; the conjecture asks for the stronger almost-sure conclusion.