Uniform convergence conjecture for weak competition in an allelopathic phytoplankton model

Let $(u,v)$ solve the reaction-diffusion system of allelopathic phytoplankton with fear function $k(x)$, subject to the parameter restrictions

$$m>1-ac-\mathbf{k}, \quad 0<c<1, \quad 0<\mathbf{k}<\frac{1}{a}-1,$$

where $\mathbf{k}$ is either $\mathbf{\widehat{k}}$ or $\mathbf{\widetilde{k}}$, and let $(u^*,v^*)$ denote the spatially homogeneous state. Uniform convergence conjecture. For any positive initial data $[u_0(x),v_0(x)]$, the solution $(u,v)$ converges uniformly to $(u^*,v^*)$ as $t\to\infty$.

This conjecture is motivated by theorems and numerical simulations for the weak competition case. The supplied text does not establish whether the asserted convergence is known beyond those results.