Gibbons' one-dimensional symmetry conjecture for uniformly convergent solutions

Let $u:\mathbb{R}^n\to[-1,1]$ be a solution of the Allen–Cahn equation in the whole of $\mathbb{R}^n$ such that

$$\lim_{x_n\to\pm\infty}u(x',x_n)=\pm1$$

uniformly with respect to $x'=(x_1,\ldots,x_{n-1})\in\mathbb{R}^{n-1}$. Gibbons' conjecture. Is $u(x)=u_0(x_n)$ for some $u_0:\mathbb{R}\to\mathbb{R}$?

This variant replaces monotonicity by uniform convergence to the two phases at infinity. The source presents it as an independently proposed rigidity problem related to De Giorgi's conjecture; the supplied text does not establish its resolution.