Bistability conjecture for the Allee-effect fear competition system

Consider the reaction–diffusion system for the Allee effect, in both its strong and weak forms, with fear function $q(x)$. Let $\mathbf{\widehat{q}}$ and $\mathbf{\widetilde{q}}$ be the two fear functions under consideration, and suppose that

$$\Delta>0,\qquad 2A_1+A_2>0,\qquad 0<c<1.$$

Bistability conjecture. There exist initial data $[u_0(x),v_0(x)]$ such that the corresponding solution $(u,v)$ converges uniformly to $(u^*,v^*)$ as $t\to\infty$, while for some other initial data $[u_1(x),v_1(x)]$, the corresponding solution converges uniformly to the spatially homogeneous state $(0,1)$ as $t\to\infty$.

The conjecture predicts bistability between the positive equilibrium $(u^*,v^*)$ and the boundary state $(0,1)$ under the stated parameter restrictions, for both strong and weak Allee effects. The supplied passage gives numerical motivation but does not provide a resolution.