De Giorgi's conjecture in dimension five

Let $n = 5$. Consider a function $u : \mathbb{R}^n \to \mathbb{R}$. We say that $u$ is a bounded classical solution to $\Delta u + u - u^3 = 0$ if $u$ 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 $x \in \mathbb{R}^n$, where $\Delta u$ denotes the Laplacian of $u$. We say that $u$ has positive first partial derivative if $\partial_1 u(x) > 0$ for all $x \in \mathbb{R}^n$, meaning the directional derivative of $u$ in the direction of the first standard basis vector is strictly positive everywhere. We say that $u$ has hyperplane level sets if for every $y$ in the range of $u$, there exists an affine subspace $S \subset \mathbb{R}^n$ such that the level set $u^{-1}(\{y\}) = \{x \in \mathbb{R}^n : u(x) = y\}$ equals $S$ and the direction of $S$ has dimension $n - 1 = 4$. Prove that if $u$ is a bounded classical solution to $\Delta u + u - u^3 = 0$ with positive first partial derivative, then $u$ has hyperplane level sets.