De Giorgi's one-dimensionality conjecture for monotone Allen–Cahn solutions

Let $u$ be a bounded solution of the Allen–Cahn equation

$$\Delta u=u^3-u\qquad\text{in }\mathbb{R}^n.$$

Assume that $u$ is monotone in one direction, namely $u_{x_n}>0$. De Giorgi's conjecture. If $n\leq 8$, then $u$ is one-dimensional.

This conjecture is the Allen–Cahn analogue of the Bernstein problem and is motivated by the connection between stable solutions and minimal hypersurfaces. The supplied context records substantial progress in the corresponding minimal-surface regularity theory, but gives no resolution status for the conjecture itself.