Refined Fisher–KPP front expansion conjecture under a third-moment condition

Let $u_0$ satisfy

$$0\leq u_0(x)\leq 1,\qquad \lim_{x\to-\infty}u_0(x)=1,$$

and

$$\int u_0(x)x^3e^x\,\mathrm{d}x<\infty.$$

Let $\mu_t^\alpha$ be the $\alpha$-level position, let $W^\alpha$ be defined by $\omega(W^\alpha)=\alpha$, and let $C$ be a constant.

Refined Fisher–KPP expansion conjecture. One has

$$\mu_t^\alpha=2t-\frac32\ln t+C+W^\alpha-\frac{3\sqrt\pi}{\sqrt t}+g\frac{\ln t}{t}+\mathcal{O}\left(\frac1t\right),$$

for some constant $g$ which does not depend on $\alpha$.

This conjecture is motivated by a solvable Fisher–KPP model and combines the predicted universal $t^{-1/2}$ correction with a logarithmic-over-linear term. The source does not establish the expansion; its level-independence conclusion also relies on the preceding level-difference conjecture.