Long-time convergence conjecture for the Neumann finite-population Fisher–KPP equation

Let $u(x,t)$ solve the Neumann problem

$$\frac{\partial u}{\partial t}=ru(1-u)+\frac{2ak}{ax+b}\frac{\partial u}{\partial x}+k\frac{\partial^2u}{\partial x^2},$$

with

$$u_x(0,t)=0,\qquad u_x(1,t)=0,$$

and initial data $u(x,0)=u_0(x)$, where $u_0(x)>0$ for some $x$. Long-time convergence conjecture.

$$\lim_{t\to\infty}u(x,t)=1.$$

The claim describes the predicted positive steady state under Neumann boundary conditions; the supplied text presents the long-run behavior as easy to predict but gives no proof or resolution.