Global asymptotic stability conjecture for locally stable steady states in spatially heterogeneous competition models

Consider the reaction–diffusion competition model

$$\begin{cases} u_t= d \Delta u+ u(m(x)- u- c v),\\ v_t= D \Delta v +v(m(x)-b u- v), \end{cases}$$

with Neumann boundary conditions on a habitat $\Omega$, where $m(x)$ is nonconstant and $0<b,c<1$. Global asymptotic stability conjecture. Every locally stable steady state is globally asymptotically stable. This conjecture concerns whether local stability determines the global dynamics of the spatially heterogeneous competition model. The source states that it was proposed and partially verified; it does not provide enough detail here to characterize the remaining cases.