De Giorgi's conjecture for dimensions at most eight

Let $n$ be a positive integer with $n \leq 8$. Let $u : \mathbb{R}^n \to \mathbb{R}$ be a function. 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 $x \in \mathbb{R}^n$. We say that $u$ has positive first partial derivative if $\partial_1 u(x) > 0$ for all $x \in \mathbb{R}^n$. We say that $u$ has hyperplane level sets if for every $y$ in the range of $u$, there exists an affine subspace $S \subseteq \mathbb{R}^n$ of codimension $1$ such that $\{x \in \mathbb{R}^n : u(x) = y\} = S$. 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.