De Giorgi's conjecture in dimension four

Consider the semilinear elliptic equation $\Delta u + u - u^3 = 0$ on $\mathbb{R}^n$, where $\Delta$ denotes the Laplacian. A function $u : \mathbb{R}^n \to \mathbb{R}$ is called a bounded classical solution 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 $u$ satisfies $\Delta u(x) + u(x) - u(x)^3 = 0$ at every point $x \in \mathbb{R}^n$. A bounded classical solution $u$ is said to have positive first partial derivative if $\partial_1 u(x) > 0$ for all $x \in \mathbb{R}^n$, meaning the directional derivative in the direction of the first standard basis vector is strictly positive everywhere. The solution $u$ is said to have hyperplane level sets if for every value $y$ in the range of $u$, there exists an affine subspace $S \subseteq \mathbb{R}^n$ of codimension $1$ (that is, whose direction has dimension $n-1$) such that the level set $u^{-1}(\{y\}) = \{x \in \mathbb{R}^n : u(x) = y\}$ coincides with $S$. De Giorgi's conjecture asserts that in dimension $n \leq 8$, every bounded classical solution with positive first partial derivative has hyperplane level sets. For $n = 4$, does this rigidity property hold? In other words, does every bounded classical solution $u : \mathbb{R}^4 \to \mathbb{R}$ to $\Delta u + u - u^3 = 0$ satisfying $\partial_1 u > 0$ everywhere necessarily have hyperplane level sets?