Level-difference conjecture for Fisher–KPP fronts

Let $u_0$ satisfy

$$0\leq u_0(x)\leq 1,$$ $$\limsup_{x\to\infty}\frac1x\ln\left[\int_x^{x(1+h)}u_0(y)\,\mathrm{d}y\right]\leq -1$$

for some (equivalently, all) $h>0$, and $\lim_{x\to-\infty}u_0(x)=1$. For $0<\alpha<\beta<1$, let $\mu_t^\alpha$ and $\mu_t^\beta$ be the corresponding level positions, and let $W^\alpha,W^\beta$ satisfy $\omega(W^\alpha)=\alpha$ and $\omega(W^\beta)=\beta$ for the critical travelling wave $\omega$.

Level-difference conjecture. One has

$$\mu_t^\alpha-\mu_t^\beta=W^\alpha-W^\beta+\mathcal{O}\left(\frac1t\right).$$

This would imply that every asymptotic term larger than $1/t$ in the level position has a coefficient independent of the level $\alpha$. The paper gives numerical evidence for a step initial condition, but the conjecture is not proved in the supplied text.