Uniqueness of localized quasimonochromatic breather profiles

Let $N\in\mathbb{N}$ with $N\geq 2$, and let $p\in C^2(\mathbb{R}^N)$ solve the equation referred to as $\operatorname{ReducedEqII}$ for some $\lambda_1,\lambda_2,\mu_2\in\mathbb{R}$. Assume that $p$ satisfies the asymptotic condition

$$\sup_{x'\in\mathbb{R}^{N-1}}|p(x_1,x')|\to 0\qquad\text{as }x_1\to\infty.$$

Uniqueness conjecture. There exist $\gamma,m,z\in\mathbb{R}$ such that

$$p(x)=\frac{\gamma}{\cosh(m(x_1-z))}.$$

This is an open classification problem for profile functions localized in one spatial direction. It is related to the Gibbons and de Giorgi conjectures concerning classification of monotone solutions of the Allen–Cahn equation, but the supplied text does not establish the conjecture.