Uniqueness conjecture for pulsating fronts of the competition system

Let $k>k^{\star}$. Let $\left(\hat{\varphi}_{1},\hat{\varphi}_{2}\right)$ and $\hat{c}$ be respectively the profile and the speed of a pulsating front solution $\left(\hat{u}_{1},\hat{u}_{2}\right)$ of $\left(\mathcal{P}\right)$. A pulsating front solution has a profile, speed, and spatially periodic traveling-wave structure as described in the preceding existence hypothesis. Uniqueness conjecture. Then $\hat{c}=c_{k}$ and there exists $\hat{\xi}\in\mathbb{R}$ such that $\left(\hat{\varphi}_{1},\hat{\varphi}_{2}\right)$ coincides with

$$\left(\xi,x\right)\mapsto\left(\varphi_{1,k},\varphi_{2,k}\right)\left(\xi-\hat{\xi},x\right).$$

This asserts uniqueness of the pulsating front speed and profile up to translation in the traveling coordinate.