Extinction and convergence conjecture for stochastic Lotka–Volterra food chains

Let $a_{11}>0$, let $\Sigma$ be positive definite, and let the initial condition satisfy $\mathbf{X}(0)=\mathbf{x}\in\mathbb{R}_+^{n,\circ}$. Suppose there is an index $j^*<n$ such that $\tilde\kappa(j^*)>0$ and $\tilde\kappa(j^*+1)<0$.

Extinction and convergence conjecture. The predators $(X_{j^*+1},\dots,X_n)$ go extinct, in the sense that

$$\mathbb{P}_x\left\{\lim_{t\to\infty}\frac{\ln X_k(t)}{t}=\tilde a_{k0}\right\}=1,\qquad k>j^*.$$

At the same time, the normalized occupation measure of $(X_1,\dots,X_{j^*})$ converges weakly to the unique invariant probability measure $\pi^{(j^*)}$ on $\mathbb{R}_+^{(j^*),\circ}$.

The conjecture would strengthen the paper's extinction results for dimensions greater than two, where only weak extinction is established. It describes both the asymptotic extinction rates of the higher predators and the limiting distribution of the surviving trophic levels.