Monotone-profile classification for localized quasimonochromatic breathers

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

$$\partial_1p(x)x_1<0$$

for every $x\in\mathbb{R}^N$ with $x_1\neq 0$.

Monotone-profile conjecture. There exist $\gamma,m>0$ such that

$$p(x)=\frac{\gamma}{\cosh(mx_1)}.$$

This is a stronger monotonicity-based classification statement in the same breather-profile setting. The supplied text gives no resolution, and the underlying reduced equation is not reproduced here.