Berestycki–Lin–Wei–Zhao's Gibbons-type conjecture for an elliptic system

Consider the elliptic system

$$\begin{cases} -\Delta u=-uv^2 & \text{in }\mathbb{R}^N,\\ -\Delta v=-u^2v & \text{in }\mathbb{R}^N,\\ u,v>0 & \text{in }\mathbb{R}^N. \end{cases}$$

Let $N\geq 2$, and let $(u,v)$ be a solution satisfying

\begin{split} \lim_{x_N\to-\infty}u(x',x_N)&=0, &\lim_{x_N\to+\infty}u(x',x_N)&=+\infty,\\ \lim_{x_N\to-\infty}v(x',x_N)&=+\infty, &\lim_{x_N\to+\infty}v(x',x_N)&=0, \end{split}

with the limits uniform in $x'\in\mathbb{R}^{N-1}$. Berestycki–Lin–Wei–Zhao's Gibbons-type conjecture. Then $(u,v)$ is $1$-dimensional. The claim is a symmetry statement for entire positive solutions of the Bose–Einstein condensation system, analogous to the Gibbons conjecture for the Allen–Cahn equation. The paper's abstract states that this conjecture is proved, so the conjecture is resolved in the cited work.