Smaller refuge with overcrowding conjecture

Consider the three-species food-chain model given by equations (1.3)–(1.1), with parameters satisfying $c < \frac{w_3}{D_3}$ and an initial condition $(u_0,v_0,r_0)$ for which the invasive species $r$ blows up in finite time in the unmodified model. There are a prey-refuge patch $\Omega_2\subset\Omega$ and an overcrowding coefficient $d_4$; the modified model includes the self-diffusion term $d_4(r^2)_{xx}$.

Smaller-refuge-with-overcrowding conjecture. There exist a patch $\Omega_2$ and a coefficient $d_4$ such that every single refuge patch of measure at least $|\Omega_2|$, with the same parameter set and initial condition, gives globally existing solutions for the modified model. In particular, $r$ does not blow up in finite time. Moreover, $\Omega_2\subset\Omega_1$, where $\Omega_1$ is the patch from the preceding conjecture.

The claim expresses the expectation that overcrowding-induced self-diffusion can prevent blow-up with a smaller refuge than refuge alone. The source gives no proof or resolution.