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on De Giorgi's conjecture in dimension six

The statement

Let $n = 6$ . Consider a function $u : \mathbb{R}^n \to \mathbb{R}$ that is a bounded classical solution to the Allen–Cahn equation $\Delta u + u - u^3 = 0$ , meaning that $u$ is twice continuously differentiable, there exists a constant $C$ such that $|u(x)| \leq C$ for all $x \in \mathbb{R}^n$ , and $\Delta u(x) + u(x) - u(x)^3 = 0$ for all $x \in \mathbb{R}^n$ . Assume furthermore that the first partial derivative of $u$ in the direction of the first coordinate is strictly positive everywhere, that is, $\partial_1 u(x) > 0$ for all $x \in \mathbb{R}^n$ . Does it follow that the level sets of $u$ are hyperplanes? In other words, for every $y$ in the range of $u$ , does there exist an affine subspace $S \subset \mathbb{R}^n$ of codimension $1$ such that $\{x \in \mathbb{R}^n : u(x) = y\} = S$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.