Gibbons-type conjecture for abelian Higgs critical points

Let $(u,\nabla)$ be an entire critical point on $\mathbb{R}^n$, and let $e_\varepsilon(u,\nabla)$ denote its energy density. Assume

$$\lim_{R\to\infty}\frac{1}{|B_R^{n-2}|}\int_{B_R^n}e_\varepsilon(u,\nabla)=2\pi.$$

Write $x=(y,z)\in\mathbb{R}^2\times\mathbb{R}^{n-2}$ and also assume

$$\lim_{|y|\to\infty}|u(y,z)|=1, \quad\text{uniformly in }z.$$

Gibbons-type conjecture. The critical point is necessarily two-dimensional: it is the pullback under the projection $\mathbb{R}^n\to\mathbb{R}^2$ of the standard degree $\pm1$ solution in $\mathbb{R}^2$, up to translation and gauge change. The excerpt presents this as a variant of the Gibbons conjecture and says that the available techniques do not currently establish the conclusion.