Full classification of Fisher-KPP solutions by exponential decay

Assume that $f$ satisfies the Fisher–KPP hypotheses. Let $u$ be a solution with $0<u<1$, and define, when the limit exists,

$$\lambda=-\lim_{x\to+\infty}\frac{\ln u(0,x)}{x}.$$

A transition front connects $0$ and $1$ if it has the front behavior described for the Fisher–KPP equation; $c^*$ is the minimal traveling-wave speed, and $c_-$ and $c_+$ are its asymptotic past and future speeds.

Full decay-characterization conjecture. For any solution $0<u<1$, the limit $\lambda$ exists in $[0,\sqrt{f'(0)}]$, and $u$ is a transition front connecting $0$ and $1$ if and only if $\lambda>0$. Furthermore, if $\lambda>0$, then $u$ has asymptotic past and future speeds $c_-$ and $c_+$ satisfying the stated speed relations with $c^*\le c_-$, and the convergence to the limiting profiles holds as in the preceding asymptotic-profile statement.

The theorem in the paper proves this characterization under an additional backward-decay hypothesis. The conjecture removes that hypothesis and extends the classification to all solutions with $0<u<1$.