Asymptotic speeds and profiles for all Fisher-KPP transition fronts

Assume that $f$ satisfies the hypotheses of the Fisher–KPP equation, and let $u$ be a transition front connecting $0$ and $1$. Let $X(t)$ be a front position and let $c^*$ be the minimal traveling-wave speed. The asymptotic past and future speeds are the limits $c_- = \lim_{t\to-\infty}X(t)/t$ and $c_+ = \lim_{t\to+\infty}X(t)/t$, when they exist; $\varphi_{c_\pm}$ denote the corresponding traveling-wave profiles.

Asymptotic-speed and profile conjecture. Any such transition front has asymptotic past and future speeds $c_\pm$ satisfying

$$c^*\le c_-\le c_+<+\infty,$$

and there is a bounded function $\xi:\mathbb{R}\to\mathbb{R}$ such that

$$u(t,X(t)+\xi(t)+\cdot)\to\varphi_{c_\pm}\quad\text{in }C^2(\mathbb{R})\quad\text{as }t\to\pm\infty.$$

The theorem preceding this conjecture establishes the result when $\liminf_{t\to-\infty}X(t)/t>c^*$; the conjecture concerns the critical case where that liminf equals $c^*$.