Conjecture on transience of the randomly switched predator-prey process

Let $X_t=(x_t,y_t)$ be the process defined by the Lotka–Volterra PDMP in the paper, with initial condition $X_0=(x_0,y_0)\in\mathbb{R}_{++}^2$. Here, $\mathbb{R}_{++}^2$ denotes the positive quadrant. Transience conjecture. Almost surely,

$$\lim_{t\to\infty}\left(x_t+y_t+\frac{1}{x_t}+\frac{1}{y_t}\right)=\infty,$$

and the process $X_t$ is transient. The conjecture concerns the long-term behavior of the randomly switched predator-prey system; the cited simulations suggest this behavior, but the paper presents it as a direction for future research and does not establish it.