Weak convergence conjecture for the surviving species in the stochastic Rosenzweig–MacArthur model

Let $M_+$ be the interior state space of the stochastic Rosenzweig–MacArthur model, let $x_1^x(t)$ denote the first component of the solution started from $x\in M_+$, and let $\gamma_{\varepsilon,\kappa}$ be the probability measure appearing in Theorem. Weak convergence conjecture. In the setting of that theorem, for every $x\in M_+$, the law of $x_1^x(t)$ converges weakly to $\gamma_{\varepsilon,\kappa}(\mathrm dx)$ as $t\to\infty$. The preceding proposition establishes convergence of the joint empirical occupation measure to a measure supported on the extinction set of the second species, but the source notes that there is no evidence that the law of the surviving species itself converges to $\gamma_{\varepsilon,\kappa}$; the claim is therefore presented as an unresolved conjecture.