Conjecture on oscillations and coexistence in the two-sex predator-prey branching process

Let $\{(Z_n,\widetilde{Z}_n)\}_{n\in\mathbb{N}_0}$ be a PP-2SDDBPO, with total population variables $T_n$ and $\widetilde{T}_n$, parameter $\mu$, and positive initial values $i,j>0$. Oscillation and coexistence conjecture. For any initial values $i,j>0$,

$$P_{(i,j)}\left(\left\{\liminf_{n\to\infty}\frac{\widetilde{T}_n}{T_n}\leq \mu<\limsup_{n\to\infty}\frac{\widetilde{T}_n}{T_n}\right\}\cap\{T_n\to\infty,\widetilde{T}_n\to\infty\}\right)=0.$$

Moreover, if $\rho_2\alpha m>\tilde{\rho}_2\tilde{\alpha}\widetilde{m}$, then

$$P_{(i,j)}\left(\left\{\liminf_{n\to\infty}\frac{\widetilde{T}_n}{T_n}>\mu\right\}\cap\{T_n\to\infty,\widetilde{T}_n\to\infty\}\right)=0,$$

and in particular

$$P_{(i,j)}(T_n\to\infty,\widetilde{T}_n\to\infty)=0.$$

The conjecture formalizes simulation-based expectations that the oscillation phase cannot persist indefinitely and that, when the predator growth parameter exceeds the prey growth parameter, coexistence has probability zero.