Optimal-diffusion conjecture for the zero-Dirichlet river competition model

Consider the two-species river competition system with diffusion rates $d,D>0$, common advection rate $\alpha>0$, growth rate $r(x)$, upstream flux boundary conditions, and zero Dirichlet conditions at $x=L$. Let $(0,v^*)$ denote its semi-trivial steady state when it exists. Optimal-diffusion conjecture. There exists $D^*=D^*(\alpha)>0$ such that, if $D=D^*$ and $d\ne D^*$, then $(0,v^*)$, whenever it exists, is globally stable. This conjecture proposes a unique evolutionarily optimal diffusion rate in the model with a lethal downstream boundary.